International Journal of Analysis and Applications Volume 16, Number 4 (2018), 484-502 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-484 A STUDY OF NON-ASSOCIATIVE ORDERED SEMIGROUPS IN TERMS OF SEMILATTICES VIA SMALLEST (DOUBLE-FRAMED SOFT) IDEALS FAISAL YOUSUFZAI1, TAUSEEF ASIF2, ASGHAR KHAN2,∗ AND BIJAN DAVVAZ3 1Military College of Engineering National University of Sciences and Technology (NUST), Islamabad, Pakistan 2Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan 3Department of Mathematics, Yazd University, Yazd, Iran ∗Corresponding author: asghar@awkum.edu.pk Abstract. Soft set theory, introduced by Molodtsov has been considered as a successful mathematical tool for modeling uncertainties. A double-framed soft set is a generalization of a soft set, consisting of union soft sets and intersectional soft sets. An ordered AG-groupoid can be referred to as a non-associative ordered semigroup, as the main difference between an ordered semigroup and an ordered AG-groupoid is the switching of an associative law. In this paper, we define the smallest left (right) ideals in an ordered AG-groupoid and use them to characterize a strongly regular class of a unitary ordered AG-groupoid along with its semilattices and double-framed soft (briefly DFS) l-ideals (r-ideals). We also give the concept of an ordered A*G**-groupoid and investigate its structural properties by using the generated ideals and DFS l-ideals (r-ideals). These concepts will verify the existing characterizations and will help in achieving more generalized results in future works. Received 2018-01-16; accepted 2018-04-06; published 2018-07-02. 2010 Mathematics Subject Classification. 00A00. Key words and phrases. DFS-sets; ordered AG-groupoid, pseudo-inverses; ordered A*G**-groupoid, left invertive law; smallest ideals and DFS ideals. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 484 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-484 Int. J. Anal. Appl. 16 (4) (2018) 485 1. Introduction The concept of soft set theory was introduced by Molodtsov in [17]. This theory can be used as a generic mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields [1, 2, 5–9]. At present, the research work on soft set theory in algebraic fields is progressing rapidly [20, 22–24]. A soft set is a parameterized family of subsets of the universe set. In the real world, the parameters of this family arise from the view point of fuzzy set theory. Most of the researchers of algebraic structures have worked on the fuzzy aspect of soft sets. Soft set theory is applied in the field of optimization by Kovkov in [13]. Several similarity measures have been discussed in [16], decision making problems have been studied in [22], reduction of fuzzy soft sets and its applications in decision making problems have been analyzed in [14]. The notions of soft numbers, soft derivatives, soft integrals and many more have been formulated in [15]. This concept have been used for forecasting the export and import volumes in international trade [26]. Recently, Jun et al. further extended the notion of softs set into double-framed soft sets and defined double-framed soft subalgebra of BCK/BCI algebra and studied the related properties in [8]. Jun et al. also defined the concept of a double-framed soft ideal (briefly, DFS ideal) of a BCK/BCI-algebra and gave many valuable results for this theory. In [12], Khan et al. have applied the idea of double-framed soft set to ordered semigroups and defined prime and irreducible DFS ideals of an ordered semigroup over a universe set U. Khan et al. have also characterized different classes of an ordered semigroup by using different DFS ideals. In the present paper, we apply the idea given by Jun et al. in [8], to ordered AG-groupoids. We introduce and investigate the notions of DFS l-ideals and DFS r-ideals, and study the relationship between these DFS ideals in detail. As an application of our results we get characterizations of a strongly regular class of a unitary ordered AG-groupoid (an ordered A*G**-groupoid) in terms of its semilattices, one-sided (two-sided) ideals based on DFS-sets and generated commutative monoids. 2. Preliminaries An AG-groupoid is a non-associative and a non-commutative algebraic structure lying in a grey area between a groupoid and a commutative semigroup. Commutative law is given by abc = cba in ternary operations. By putting brackets on the left of this equation, i.e. (ab)c = (cb)a, in 1972, M. A. Kazim and M. Naseeruddin introduced a new algebraic structure called a left almost semigroup abbreviated as an LA-semigroup [10]. This identity is called the left invertive law. P. V. Protic and N. Stevanovic called the same structure an Abel-Grassmann’s groupoid abbreviated as an AG-groupoid [21]. This structure is closely related to a commutative semigroup because a commutative AG-groupoid is a semigroup [18]. It was proved in [10] that an AG-groupoid S is medial, that is, ab · cd = ac · bd holds for all Int. J. Anal. Appl. 16 (4) (2018) 486 a,b,c,d ∈ S. An AG-groupoid may or may not contain a left identity. The left identity of an AG-groupoid permits the inverses of elements in the structure. If an AG-groupoid contains a left identity, then this left identity is unique [18]. In an AG-groupoid S with left identity, the paramedial law ab · cd = dc · ba holds for all a,b,c,d ∈ S. By using medial law with left identity, we get a · bc = b ·ac for all a,b,c ∈ S. We should genuinely acknowledge that much of the ground work has been done by M. A. Kazim, M. Naseeruddin, Q. Mushtaq, M. S. Kamran, P. V. Protic, N. Stevanovic, M. Khan, W. A. Dudek and R. S. Gigon. One can be referred to [3, 4, 11, 18, 19, 21, 25] in this regard. An AG-groupoid (S, ·) together with a partial order ≤ on S that is compatible with an AG-groupoid operation, meaning that for x,y,z ∈ S, x ≤ y ⇒ zx ≤ zy and xz ≤ yz, is called an ordered AG-groupoid [28]. Let us define a binary operation ”◦e” (e-sandwich operation) on an ordered AG-groupoid (S, ·,≤) with left identity e as follows: a◦e b = ae · b, ∀ a,b ∈ S. Then (S, ◦e,≤) becomes an ordered semigroup [28]. Note that an ordered AG-groupoid is the generalization of an ordered semigroup because if an ordered AG-groupoid has a right identity then it becomes an ordered semigroup. Let ∅ 6= A ⊆ S, we denote (A] by (A] := {x ∈ S/x ≤ a for some a ∈ A}. If A = {a}, then we write ({a}]. For ∅ 6= A,B ⊆ S, we denote AB =: {ab/a ∈ A,b ∈ B}. • Let S be an ordered AG-groupoid. By an ordered AG-subgroupoid of S, we means a nonempty subset A of S such that (A2] ⊆ A. • A nonempty subset A of an ordered AG-groupoid S is called a left (right) ideal of S if: (i ) SA ⊆ A (AS ⊆ A); (ii) if a ∈ A and b ∈ S such that b ≤ a, then b ∈ A. Equivalently: A nonempty subset A of an ordered AG-groupoid S is called a left (right) ideal of S if (SA] ⊆ A ((AS] ⊆ A). • By two-sided ideal or simply ideal, we mean a nonempty subset of an ordered AG-groupoid S which is both left and right ideal of S. Lemma 2.1. [28] Let S be an ordered AG-groupoid and ∅ 6= A,B ⊆ S. Then the following hold: (i) A ⊆ (A] ; (ii) If A ⊆ B, then (A] ⊆ (B] ; (iii) (A] (B] ⊆ (AB] ; (iv) (A] = ((A]] ; (vi) ((A] (B]] = (AB] ; (vii) (T] = T, for every ideal T of S; Int. J. Anal. Appl. 16 (4) (2018) 487 (viii) (SS] = S = SS, if S has a left identity. 3. Soft Sets In [24], Sezgin and Atagun introduced some new operations on soft set theory and defined soft sets in the following way. Let U be an initial universe set, E a set of parameters, P(U) the power set of U and A ⊆ E. Then a soft set fA over U is a function defined by: fA : E → P(U) such that fA(x) = ∅, if x /∈ A. Here fA is called an approximate function. A soft set over U can be represented by the set of ordered pairs fA = {(x, fA(x)) : x ∈ E, fA(x) ∈ P(U)} . It is clear that a soft set is a parameterized family of subsets of U. The set of all soft sets is denoted by S(U). • Let fA, fB ∈ S(U). Then fA is a soft subset of fB, denoted by fA ∼ ⊆ fB if fA(x) ⊆ fB(x) for all x ∈ S. Two soft sets fA, fB are said to be equal soft sets if fA ∼ ⊆ fB and ∼ fB ⊆ fA and is denoted by fA ∼ = fB. The union of fA and fB, denoted by fA ∼ ∪fB, is defined by fA ∼ ∪fB = fA∪B, where fA∪B(x) = fA(x) ∪fB(x), ∀ x ∈ E. In a similar way, we can define the intersection of fA and fB. • Let S be an ordered AG-groupoid, let fA, fB ∈ S(U). Then the soft product [24] of fA and fB, denoted by fA ∼ ◦ fB, is defined as follows: (fA ∼ ◦ fB)(x) =   ⋃ (y,z)∈Ax {fA(y) ∩gB (z)} if Ax 6= ∅ ∅ if Ax = ∅ where Ax = {(y,z) ∈ S ×S/x ≤ yz}. • A double-framed soft pair 〈 (f+A , f − A ; A 〉 is called a double-framed soft set (briefly, DFS-set of A) [8] of A over U, where f+A and f − A are mappings from A to P(U). The set of all DFS-sets of A over U will be denoted by DFS(U). • Let fA = 〈 (f+A , f − A ); A 〉 and gA = 〈 (g+A, g − A ); A 〉 be two double-framed soft sets of an ordered AG- groupoid S over U. Then the uni-int soft product [12], denoted by fA � gA = 〈 (f+A ∼ ◦ g+A, f − A ∼ ? g−A ); A 〉 is defined to be a double-framed soft set of S over U, in which f+A ∼ ◦ g+A and f − A ∼ ? g−A are mapping from S to P(U), given as follows: Int. J. Anal. Appl. 16 (4) (2018) 488 f+A ∼ ◦ g+A : S −→ P(U),x 7−→   ⋃ (y,z)∈Ax {f+A (y) ∩g + A (z)} if Ax 6= ∅ ∅ if Ax = ∅, f−A ∼ ? g−A : S −→ P(U),x 7−→   ⋂ (y,z)∈Ax {f−A (y) ∪g − A (z)} if Ax 6= ∅ U if Ax = ∅. • Let fA = 〈 (f+A , f − A ); A 〉 and gA = 〈 (g+A, g − A ); A 〉 be two double-framed soft sets over a common universe set U. Then 〈 (f+A , f − A ); A 〉 is called a double-framed soft subset (briefly, DFS-subset ) [12] of 〈 (g+A, g − A ); A 〉 , denote by 〈 (f+A , f − A ); A 〉 v 〈 (g+A, g − A ); A 〉 if: (i) A ⊆ B; (ii) (∀e ∈ A)   f+A and g+A are identical approximations (f+A (e) ⊆ g+A (e)) f−A and g − A are identical approximations (f − A (e) ⊇ g − A (e))   . • For two DFS-sets fA = 〈 (f+A , f − A ); A 〉 and gA = 〈 (g+A, g − A ); A 〉 over U are said to be equal, denoted by〈 (f+A , f − A ); A 〉 = 〈 (g+A, g − A ); A 〉 , if 〈 (f+A , f − A ); A 〉 v 〈 (g+A, g − A ); A 〉 and 〈 (g+A, g − A ); A 〉 v 〈 (f+A , f − A ); A 〉 . • For two DFS-sets fA = 〈 (f+A , f − A ); A 〉 and gA = 〈 (g+A, g − A ); A 〉 over U, the DFS int-uni set [12] of〈 (f+A , f − A ); A 〉 and 〈 (g+A, g − A ); A 〉 , is defined to be a DFS-set 〈 (f+A ∩g + A, f − A ∪g − A ); A 〉 , where f+A ∩g + A and f−A ∪g − A are mapping given as follows: f+A ∩g + A : A −→ P(U),x 7−→ f + A (x) ∩g + A (x); f−A ∪g − A : A −→ P(U),x 7−→ f − A (x) ∪g − A (x). It is denoted by 〈 (f+A , f − A ); A 〉 u 〈 (g+A, g − A ); A 〉 = 〈 (f+A ∩g + A, f − A ∪g − A ); A 〉 . • A double-framed soft set fA = 〈 (f+A , f − A ); A 〉 of S over U is called a double-framed soft AG-subgroupoid (briefly, DFS AG-subgroupoid) of S over U if it satisfies f+A (xy) ⊇ f + A (x)∩f + A (y), f − A (xy) ⊆ f − A (x)∪f − A (y), ∀ x, y ∈ S. • A double-framed soft set fA = 〈 (f+A , f − A ); A 〉 of S over U is called (i) a double-framed soft left ideal (briefly, DFS l-ideal) of S over U if it satisfies: (a) f+A (xy) ⊇ f + A (y) and f − A (xy) ⊆ f − A (y); (b) x ≤ y =⇒ f+A (x) ⊇ f + A (y) and f − A (x) ⊆ f − A (y), ∀ x, y ∈ S. (ii) a double-framed soft right ideal (briefly, DFS r-ideal) of S over U if it satisfies: (a) f+A (xy) ⊇ f + A (x) and f − A (xy) ⊆ f − A (x); (b) x ≤ y =⇒ f+A (x) ⊇ f + A (y) and f − A (x) ⊆ f − A (y), ∀ x, y ∈ S. (iii) a double-framed soft ideal (briefly, DFS ideal) of S over U, if it is both DFS l-ideal and DFS r-ideal of S over U. • Let A be a nonempty subset of S. Then the characteristic double-framed soft mapping of A, denoted by 〈 (X+A , X − A ); A 〉 = XA is defined to be a double-framed soft set, in which X+A and X − A are soft mappings Int. J. Anal. Appl. 16 (4) (2018) 489 over U, given as follows: X+A : S −→ P(U),x 7−→   U if x ∈ A∅ if x /∈ A, X−A : S −→ P(U),x 7−→   ∅ if x ∈ AU if x /∈ A. Note that the characteristic mapping of the whole set S, denoted by XS = 〈 (X+S , X − S ); S 〉 , is called the identity double-framed soft mapping, where X+S (x) = U and X − S (x) = ∅, ∀ x ∈ S. The following result holds for an ordered semigroup [6] just because of the closure property which makes very clear for an ordered AG-groupoid to hold the same Lemma. Lemma 3.1. For a nonempty subset A of an ordered AG-groupoid S, the following conditions are equivalent: (i) A is a left ideal (right ideal ) of S; (ii) The DFS set XA of S over U is a DFS l-ideal (DFS r-ideal ) of S over U. The following result holds for an ordered semigroup [12] just because of the closure property which makes very clear for an ordered AG-groupoid to hold the same Lemma. Lemma 3.2. Let fA = 〈(f+A ,f − A ); A〉 be any DFS-set of an ordered AG-groupoid S over U. Then fA is a DFS r-ideal (l-ideal) of S over U if and only if fA �XS v fA (XS �fA v fA). • A double-framed soft set fA = 〈 (f+A , f − A ); A 〉 of S over U is called DFS semiprime if fA(x) w fA(x2), ∀ x ∈ A. Lemma 3.3. Let A be any right (left) ideal of an ordered AG-groupoid S. Then A is semiprime if and only if XA is DFS semiprime. Proof. Let A be a right (left) ideal of S, then by Lemma 3.1, XA is a DFS r-ideal (DFS l-ideal) of S over U. Let a2 ∈ A, then X+A (a) ⊇ X + A (a 2), therefore X+A (a 2) = U ⊆ X+A (a), this implies X + A (a) = U and similarly X−A (a) = ∅. Thus a ∈ A and therefore A is semiprime. Converse is simple. � Remark 3.1. The set (DFS(U),�,v) forms an ordered AG-groupoid and satisfies all the basic laws. Remark 3.2. If S is an ordered AG-groupoid, then XS �XS = XS. The following result also holds for an ordered semigroup [12] just because of the closure property which is very trivial for an ordered AG-groupoid to hold the same Lemma. Lemma 3.4. Let S be an ordered AG-groupoid. For ∅ 6= A,B ⊆ S, the following assertions hold: Int. J. Anal. Appl. 16 (4) (2018) 490 (i) A ⊆ B ⇔ XA vXB; (ii) XA uXB = XA∩B; (iii) XA tXB = XA∪B; (iv) XA �XB = X(AB]. 4. On DFS strongly regular ordered AG-groupoids Throughout this paper, let E = S, where S is a unitary ordered AG-groupoid, unless otherwise stated. By a unitary ordered AG-groupoid, we shall mean an ordered AG-groupoid with left identity. 4.1. Basic Results. This section contains some examples and basic results which will be essential for up coming section. Theorem 4.1. Let S be an ordered AG-groupoid. A nonempty subset A of S is a left (resp. right) ideal of S if and only if the DFS-set 〈 (g+B, g − B ); B 〉 , defined by g+B (x) =   γ1 if x ∈ Aγ2 if x ∈ S\A   and g−B (x) =   δ1 if x ∈ Aδ2 if x ∈ S\A   , is a DFS l-ideal (resp. DFS r-ideal) of S over U, where γ1,γ2,δ1,δ2 ⊆ U such that γ2 ⊆ γ1 and δ1 ⊆ δ2. Proof. Necessity. Let x,y ∈ S be such that x ≤ y. If y /∈ A, then g+B (y) = γ2 ⊆ g + B (x) and g − B (y) = δ2 ⊇ g−B (x). If y ∈ A, then γ1 = g + B (y) and δ2 = g − B (y). Since x ≤ y ∈ A, and A is a left ideal of S, we have x ∈ A. Then g+B (x) = γ1 = g + B (y) and g − B (x) = δ1 = g − B (y). For x,y ∈ S, we discuss the following two cases. Case 1. If x ∈ S and y ∈ A, then xy ∈ A and we have g+B (x) = γ1 = g + B (y) and g − B (x) = δ1 = g − B (y). Case 2. If x ∈ S and y /∈ A, then g+B (y) = γ2 ⊆ g + B (xy) and g − B (y) = δ2 ⊇ g − B (xy). Therefore 〈 (g+B, g − B ); B 〉 is a DFS l-ideal of S over U. Similarly we can prove the result for a DFS r-ideal of S over U. Sufficiency. Assume that 〈 (g+B, g − B ); B 〉 is a DFS l-ideal of S over U. Let x,y ∈ S be such that x ≤ y. If y ∈ A, then g+B (y) ⊇ γ1 and g − B (y) ⊆ δ1. Since g + B (x) ⊇ g + B (y) ⊇ γ1 and g − B (x) ⊆ g + B (y) ⊆ δ1, we have x ∈ A. Let x ∈ S and y ∈ A, then g+B (y) = γ1 and g − B (y) = δ1. By hypothesis, g + B (xy) ⊇ g + B (y) = γ1 and g−B (xy) ⊆ g + B (y) = δ1. Hence xy ∈ A. Thus A is a left ideal of S. Similarly, we can show that A is a right ideal of S. � Example 4.1. There are six different chemicals which have been used in an experiment. Take a collection of chemicals as the initial universe set U given by U = {γ1,γ2,γ3,γ4,γ5,γ6}. Let a set of parameters E = {1, 2, 3, 4, 5} be a set of particular properties of each chemical in U with the following type of natures: Int. J. Anal. Appl. 16 (4) (2018) 491 1 stands for the parameter ”density ”, 2 stands for the parameter ”melting point ”, 3 stands for the parameter ”combustion”, 4 stands for the parameter ”enthalpy ”, 5 stands for the parameter ”toxicity ”. Let us define the following binary operation and order on a set of parameters E as follows. ∗ 1 2 3 4 5 1 2 2 4 4 5 2 2 2 2 2 5 3 1 2 3 4 5 4 1 2 1 2 5 5 1 5 5 5 5 ≤= {(1, 1), (1, 2), (3, 3), (1, 3), (4, 4), (1, 5), (5, 5), (2, 2)}. It is easy to observe that (E,∗,≤) is a unitary ordered AG-groupoid. Let A = {1, 2, 5} and define a DFS-set 〈 (f+A , f − A ); A 〉 of S over U as follows: f+A (x) =   {γ1,γ2} if x = 1 {γ1,γ2,γ3} if x = 2 {γ5} if x = 3 {γ5} if x = 4 {γ1,γ2,γ3,γ4} if x = 5   and f−A (x) =   {γ1,γ2,γ3} if x = 1 {γ1,γ2} if x = 2 { } if x = 3 { } if x = 4 {γ2} if x = 5   . Then it is easy to verify that 〈 (f+A , f − A ); A 〉 is a DFS l-ideal of S over U. Let B = {1, 2, 4} and define a DFS-set 〈 (g+B, g − B ); B 〉 of S over U as follows: g+B (x) =   {γ4} if x = 1 {γ2,γ3,γ4} if x = 2 {γ1,γ2,γ5} if x = 3 {γ1,γ2,γ5} if x = 4 {γ2,γ4} if x = 5   and g−B (x) =   {γ1,γ2,γ3,γ4} if x = 1 {γ2,γ3} if x = 2 {γ1,γ2, ...,γ5} if x = 3 {γ1,γ2,γ3,γ4} if x = 4 {γ2,γ3} if x = 5   . Then it is easy to verify that 〈 (g+B, g − B ); B 〉 is a DFS r-ideal of S over U. Remark 4.1. Every DFS r-ideal of a unitary ordered AG-groupoid S over U is a DFS l-ideal of S over U but the converse inclusion is not true in general which can be followed from above example. Int. J. Anal. Appl. 16 (4) (2018) 492 Lemma 4.1. Let R be a right ideal and L be a left ideal of a unitary ordered AG-groupoid S. Then (RL] is a left ideal of S. Proof. Let R be a right ideal and L be a left ideal of S. Then by using Lemma 2.1, we get S(RL] = (SS](RL] ⊆ (SS ·RL] = (SR ·SL] ⊆ (SR · (SL]] = (SR ·L] = ((SS]R ·L] ⊆ ((SS)R ·L] = ((RS)S ·L] ⊆ ((RS]S ·L] ⊆ (RL],which shows that (RL] is a left ideal of S. � Lemma 4.2. Let S be a unitary ordered AG-groupoid. If a = a2 for all a ∈ S, then Ra = (Sa∪Sa2] is the smallest right ideal of S containing a. Proof. Assume that a = a2 for all a ∈ S. Then by using Lemma 2.1, we have (Sa∪Sa2]S = (Sa∪Sa2](S] ⊆ ((Sa∪Sa2)S] = (Sa ·S ∪Sa2 ·S] = (Sa ·SS ∪Sa2 ·SS] = (S ·aS ∪S ·a2S] = (a ·SS ∪a2 ·SS] = (a2 ·SS ∪a2 ·SS] = (SS ·a2 ∪SS ·a2] = (Sa∪Sa2], which shows that (Sa∪Sa2] is a right ideal of S. It is easy to see that a ∈ (Sa∪Sa2]. Let R be another right ideal of S containing a. Since (Sa∪Sa2] = (SS ·a∪a ·Sa] = (aS ·S ∪a ·Sa] ⊆ (RS ·S ∪RS] ⊆ R, Hence (Sa∪Sa2] is the smallest right ideal of S containing a. � Lemma 4.3. Let S be a unitary ordered AG-groupoid and a = a2 for all a ∈ S. Then S becomes a commu- tative monoid. Proof. It is simple. � Corollary 4.1. Ra = (Sa∪Sa2] is the smallest right ideal of an ordered commutative monoid S containing a. Lemma 4.4. Let S be a unitary ordered AG-groupoid and a ∈ S. Then La = (Sa] is the smallest left ideal of S containing a. Proof. It is simple. � Theorem 4.2. Let S be a unitary ordered AG-groupoid and ∅ 6= E ⊆ S . Then the following assertions hold: (i) E forms a semilattice, where E = {x ∈ S : x = x2}; (ii) E is a singleton set, if a = ax ·a, ∀ a,x ∈ S. Int. J. Anal. Appl. 16 (4) (2018) 493 Proof. (i). It is simple. (ii). Let y,z ∈ E. Then by using (i), we get y = yz ·y = zy ·y = yy ·z = yz = zy = zz ·y = yz ·z = zy ·z = z. � • Recall that an ordered AG**-groupoid is an ordered AG-groupoid in which a · bc = b ·ac, ∀ a,b,c ∈ S. Note that an ordered AG**-groupoid also satisfies the paramedial law as well. Now let us introduce the concept of an ordered A*G**-groupoid as follows: • An ordered AG**-groupoid S is called an ordered A*G**-groupoid if S = (S2]. Corollary 4.2. Let S be an ordered A*G**-groupoid and ∅ 6= E ⊆ S . Then the following assertions hold: (i) E forms a semilattice, where E = {x ∈ S : x = x2}; (ii) E is a singleton set if a = ax ·a, ∀ a,x ∈ S. Lemma 4.5. Let S be an ordered A*G**-groupoid. Then 〈R〉a2 = (Sa 2 ∪a2] (〈L〉a = (Sa∪a]) is the right (resp. left) ideal of S. Proof. Let a ∈ S, then by using Lemma 2.1, we get (Sa2 ∪a2]S = (Sa2 ∪a2](S] = ((Sa2 ∪a2)S] = (Sa2 ·S ∪a2S] = (SS ·a2S ∪SS ·aa] = (S ·a2S ∪Sa2] = (a2 ·SS ∪Sa2] = (Sa2] ⊆ (Sa2 ∪a2], which is what we set out to prove. Similarly we can prove that S(Sa∪a] ⊆ (Sa∪a]. � • An element a of an ordered AG-groupoid S is called a strongly regular element of S, if there exists some x in S such that a ≤ ax ·a and ax = xa, where x is called a pseudo-inverse of a. S is called strongly regular ordered AG-groupoid if all elements of S are strongly regular. Theorem 4.3. Let S be an ordered AG-groupoid (an ordered A*G**-groupoid) with left identity. An element a of S is strongly regular if and only if a ≤ ax ·ay for some x,y ∈ S. Proof. Necessity. Let a ∈ S is strongly regular, then a ≤ ax ·a ≤ (ax) · (xa)(ax ·a) = (ax) · (a ·ax)(ax) = (ax) ·a((a ·ax)x) = ax ·ay, where (a ·ax)x = y ∈ S. Thus a ≤ ax ·ay for some x,y ∈ S. Sufficiency. Let a ∈ S such that a ≤ ax·ay for some x,y ∈ S, then a ≤ ax·ay = (ay·x)a = (xy·a)a = ua·a, where xy = u ∈ S. Thus au ≤ (ua ·a)u = ua ·ua = u(ua ·a) ≤ ua, and a ≤ ua ·a = au ·a. Thus S is strongly regular. � Int. J. Anal. Appl. 16 (4) (2018) 494 Lemma 4.6. Let fA = 〈 (f+A , f − A ); A 〉 be any DFS r-ideal (DFS l-ideal) of a strongly regular ordered A*G**-groupoid S over U. Then the following assertions hold: (i) fA = fA �XS (fA = XS �fA); (ii) fA is DFS semiprime. Proof. It is simple. � 4.2. Characterization Problems. In this section, we generalize the results of an ordered semigroup and get some interesting characterizations which we usually do not find in an ordered semigroup. From now onward, R (resp. L) will denote any right (resp. left) ideal of an ordered AG-groupoid S; Ra (resp. La) will denote any smallest right (resp. smallest left) ideal of S containing a. Any DFS r-ideal of an ordered AG-groupoid S (resp. DFS l-ideal of S) over U will be denoted by fA (resp. gB) unless otherwise specified. Theorem 4.4. Let fA,gB be any DFS l-ideals of a unitary ordered AG-groupoid S. Then the following conditions are equivalent: (i) S is strongly regular; (ii) S is strongly regular commutative monoid; (iii) (RaLa] ∩La = ((Ra ·RaLa)La ·La], (a = a2, ∀ a ∈ S); (iv) (RL] ∩L = ((R ·RL)L ·L]; (v) fA ugB = (fA �gB) �fA; (vi) S is strongly regular and |E|= 1, (a = ax ·a, ∀ a,x ∈ E); (vii) S is strongly regular and ∅ 6= E ⊆ S is semilattice. Proof. (i) =⇒ (vii) : It can be followed from Theorem 4.2 (i). (vii) =⇒ (vi) : It can be followed from Theorem 4.2 (ii). (vi) =⇒ (v) : Let fA and gB be any DFS l-ideals of a strongly regular S over U. Now for a ∈ S, there exist some x,y ∈ S such that a ≤ ax·ay = ya·xa ≤ y(ax·ay)·xa = (ax)(y·ay)·xa = (ay·y)(xa)·xa = (y2a·xa)(xa). Thus (y2a ·xa,xa) ∈ Aa. Therefore ((f+A ∼ ◦ g+B ) ∼ ◦ f+A )(a) = ⋃ (y2a·xa,xa)∈Aa { (f+A ∼ ◦ g+B )(y 2a ·xa) ∩f+A (xa) } ⊇ ⋃ y2a·xa≤y2a·xa {f+A (y 2a) ∩g+B (xa)}∩f + A (xa) ⊇ f+A (y 2a) ∩g+B (xa) ∩f + A (xa) ⊇ f + A (a) ∩g + B (a), and similarly, we get Int. J. Anal. Appl. 16 (4) (2018) 495 ((f−A ∼ ? g−B ) ∼ ? f−A )(a) = ⋂ (y2a·xa,xa)∈Aa { (f−A ∼ ? g−B )(y 2a ·xa) ∪f−A (xa) } ⊆ ⋂ y2a·xa≤y2a·xa {f−A (y 2a) ∪g−B (xa)}∪f − A (xa) ⊆ f−A (y 2a) ∪g−B (xa) ∪f − A (xa) ⊆ f − A (a) ∪g − B (a), which shows that (fA � gB) � fA w fA u gB. By using Lemmas 3.2 and 4.6, it is easy to show that (fA �gB) �fA v fA ugB. Thus fA ugB = (fA �gB) �fA. (v) =⇒ (iv) : Let R and L be any right and left ideals of S respectively. Then by using Lemmas 3.1 and 4.1, X(RL] and XL are the DFS l-ideals of S over U. Now by using Lemma 3.4, we get X(RL]∩L = X(RL] uXL = (X(RL] �XL) �X(RL] = X((RL]L·(RL]], which give us (RL] ∩L = ((RL]L · (RL]]. Now by using Lemma 2.1, we get ((RL]L · (RL]] = ((RL)L ·RL] = (L2R ·RL] = (LR ·RL2] = (R(LR ·L2)] = (R(L2 ·RL)] = (R(R ·L2L)] = (R ·RL3] = (R(R ·L2L)] = (R(L2 ·RL)] = ((R ·RL)L ·L]. (iv) =⇒ (iii) : It is simple. (iii) =⇒ (ii) : Since (Sa∪Sa2] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a, where a = a2, ∀ a ∈ S. Thus by using given assumption and Lemma 2.1, we get a ∈ ((Sa∪Sa2](Sa]] ∩ (Sa] = (((Sa∪Sa2] · (Sa∪Sa2](Sa])(Sa] · (Sa]] = (((Sa∪Sa2) · (Sa∪Sa2)(Sa))(Sa) · (Sa)] ⊆ (S(Sa) · (Sa)] = (S2a ·Sa] = (Sa ·Sa] = (aS ·aS]. Hence by using Lemma 4.2, S is strongly regular commutative monoid. (ii) =⇒ (i) : It is obvious. � Theorem 4.5. Let S be an ordered AG-groupoid. Then the following conditions are equivalent: (i) S is strongly regular ; (ii) S is strongly regular commutative monoid ; (iii) Ra ∩La = (Ra(LaRa ·Ra)], (a = a2, ∀ a ∈ S); (iv) R∩L = (R(LR ·R)]; (v) fA ugB = f3A �gB; Int. J. Anal. Appl. 16 (4) (2018) 496 (vi) S is strongly regular and |E|= 1, (a = ax ·a, ∀ a,x ∈ E); (vii) S is strongly regular and ∅ 6= E ⊆ S is semilattice. Proof. (i) =⇒ (vii) : It can be followed from Theorem 4.2 (i). (vii) =⇒ (vi) : It can be followed from Theorem 4.2 (ii). (vi) =⇒ (v) : Let fA and gB be any DFS r-ideal and DFS l-ideal of a strongly regular S over U respectively. From Lemma 3.2, it is easy to show that f+3A �g + B v f + A ug + B . Now for a ∈ S, there exist some x,y ∈ S such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = y(ax ·ay) ·x(ax ·ay) = (ax)(y ·ay) · (ax)(x ·ay) = (ax)(ay2) · (ax)(a ·xy) = (y2a)(xa) · (ax)(a ·xy) = ((ax)(a ·xy))(xa) ·y2a = ((ax)(a ·xy))(ex ·a) ·y2a = ((ax)(a ·xy))(ax ·e) ·y2a = bc ·y2a = d ·y2a, where d = bc = ((ax)(a ·xy))(ax ·e). Thus ((f+A ∼ ◦ f+A ) ∼ ◦ f+A )(d) = ⋃ d≤bc {(f+A ∼ ◦ f+A )(b) ∩f + A (c)}⊇ (f + A ∼ ◦ f+A )(b) ∩f + A (c) = ⋃ b≤(ax)(a·xy) {f+A (ax) ∩f + A (a ·xy)}∩f + A (ax ·e) ⊇ f+A (ax) ∩f + A (a ·xy) ∩f + A (ax ·e) ⊇ f + A (a). Therefore (f+3A ∼ ◦ g+B )(a) = ⋃ a≤d·y2a {((f+A ∼ ◦ f+A ) ∼ ◦ f+A )(d) ∩g + B (y 2a)}⊇ f+A (a) ∩g + B (a), which shows that f+A ∩g + B ⊆ f +3 A ∼ ◦ g+B , and similarly f − A ∪g − B ⊇ f −3 A ∼ ? g−B . Thus fA ugB = f 3 A �gB. (v) =⇒ (iv): Let R and L be any right and left ideals of S respectively. Then by using Lemma 3.1, XR and XL are the DFS r-ideal and DFS l-ideal of S over U respectively. Now by using Lemma 3.4, we get XR∩L = XR uXL = ((XR �XR) �XR) �XL = X(R3] �XL = X((R3]L], which implies that R ∩ L = ((R3]L]. Now by using Lemma 2.1, we get R ∩ L = ((R3]L] = (R3L] = (R2R ·L] = (LR ·R2] = (R2 ·RL] = (R ·R2L] = (R(LR ·R)]. (iv) =⇒ (iii) : It is simple. Int. J. Anal. Appl. 16 (4) (2018) 497 (iii) =⇒ (ii) : Since (Sa∪Sa2] is the smallest right ideal of S containing a and (Sa] is the smallest left ideal of S containing a. Thus by using given assumption and Lemma 2.1, we get a ∈ (Sa∪Sa2] ∩ (Sa] = ((Sa∪Sa2]((Sa](Sa∪Sa2] · (Sa∪Sa2])] = ((Sa∪Sa2)((Sa)(Sa∪Sa2) · (Sa∪Sa2))] ⊆ (S(S(Sa∪Sa2) · (Sa∪Sa2))] = (S((S2a∪S2a2)(Sa∪Sa2))] = ((S2a∪S2a2)(S(Sa∪Sa2))] = ((S2a∪S2a2)(S2a∪S2a2)] = ((Sa∪a2S2)(Sa∪a2S2)] = ((Sa∪S2a ·a)(Sa∪S2a ·a)] ⊆ ((Sa∪Sa)(Sa∪Sa)] = (Sa ·Sa] = (aS ·aS]. Hence by using Lemma 4.2, S is strongly regular commutative monoid. (ii) =⇒ (i) : It is obvious. � Let S be an ordered A*G**-groupoid. From now onward, R (resp. L) will denote any right (resp. left) ideal of S; 〈R〉a2 will denote a right ideal (Sa 2 ∪ a2] of S containing a2 and 〈L〉a will denote a left ideal (Sa∪a] of S containing a; fA (resp. gB) will denote any DFS r-ideal over U (resp. DFS l-ideal over U) of S unless otherwise specified. Theorem 4.6. Let S be an ordered A*G**-groupoid. Then S is strongly regular if and only if 〈R〉a2 ∩〈L〉a = (〈R〉2a2 〈L〉 2 a] and 〈R〉a2 is semiprime. Proof. Necessity: Let S be strongly regular. It is easy to see that (〈R〉2a2 〈L〉 2 a] ⊆ 〈R〉a2 ∩ 〈L〉a . Let a ∈ 〈R〉a2 ∩〈L〉a . Then there exist some x,y ∈ S such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = (x ·ay)(ax) · (y ·ay)(ax) = (a ·xy)(ax) · (ay2)(ax) = (a ·xy)(ax) · (xa)(y2a) ∈ (〈R〉a2 S · 〈R〉a2 S)(S 〈L〉a ·S 〈L〉a) ⊆〈R〉 2 a2 〈L〉 2 a , which shows that 〈R〉a2 ∩〈L〉a = (〈R〉 2 a2 〈L〉 2 a]. It is easy to see that 〈R〉a2 is semiprime. Int. J. Anal. Appl. 16 (4) (2018) 498 Sufficiency: Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a respectively. Thus by using given assumption and Lemma 2.1, we get a ∈ (Sa2 ∪a2] ∩ (Sa∪a] = ((Sa2 ∪a2]2(Sa∪a]2] = ((Sa2 ∪a2)(Sa2 ∪a) · (Sa∪a)(Sa∪a)] ⊆ (S(Sa2 ∪a) ·S(Sa∪a)] = ((S ·Sa2 ∪Sa)(S ·Sa∪Sa)] = ((a2S ·S ∪Sa)(aS ·S ∪Sa)] = ((a2S ·S ∪Sa)(aS ·S ∪Sa)] = ((Sa2 ∪Sa)(Sa∪Sa)] = ((a2S ∪Sa)(Sa∪Sa)] = ((Sa ·a∪Sa)(Sa∪Sa)] ⊆ ((Sa∪Sa)(Sa∪Sa)] = (Sa ·Sa] = (aS ·aS]. This implies that S is strongly regular. � Corollary 4.3. Let S be an ordered A*G**-groupoid. Then S is strongly regular if and only if 〈R〉a2∩〈L〉a = (〈L〉2a 〈R〉 2 a2 ] and 〈R〉a2 is semiprime. Theorem 4.7. Let S be an ordered A*G**-groupoid. Then the following conditions are equivalent: (i) S is strongly regular ; (ii) 〈R〉a2 ∩〈L〉a = (〈L〉 2 a 〈R〉 2 a2 ] and 〈R〉a2 is semiprime; (iii) R∩L = (L2R2] and R semiprime; (iv) fA ugB = (fA �gB) � (fA �gB) and fA is DFS semiprime; (v) S is strongly regular and |E|= 1, (a = ax ·a, ∀ a,x ∈ E); (vi) S is strongly regular and ∅ 6= E ⊆ S is semilattice. Proof. (i) =⇒ (vi) : It can be followed from Corollary 4.2 (i). (vi) =⇒ (v) : It can be followed from Corollary 4.2 (ii). (v) =⇒ (iv) : Let fA and gB be any DFS r-ideal and DFS l-ideal of a strongly regular S over U respectively. From Lemma 3.2, it is easy to show that (fA �gB) � (fA �gB) v fA ugB. Now for a ∈ S, there exist some x,y ∈ S such that a ≤ ax ·ay ≤ (ax ·ay)x · (ax ·ay)y = (ax ·ay) · ((ax ·ay)x)y = (ax ·ay) · (yx)(ax ·ay) = (ax ·ay) · (ax)(yx ·ay) = (ax ·ay) · (ay ·yx)(xa) = (ax ·ay) · ((yx ·y)a)(xa) = (ax)((yx ·y)a) · (ay)(xa) = (ax)(ba) · (ay)(xa), where yx ·y = b. Int. J. Anal. Appl. 16 (4) (2018) 499 Thus (ax · ba,ay ·xa) ∈ Aa. Therefore ((f+A ∼ ◦ g+B ) ∼ ◦ (f+A ∼ ◦ g+B ))(a) = ⋃ (ax·ba,ay·xa)∈Aa {(f+A ∼ ◦ g+B )(ax · ba) ∩ (f + A ∼ ◦ g+B )(ay ·xa)} ⊇ ⋃ ax·ba≤ax·ba {f+A (ax) ∩g + B (ba)}∩ ⋃ ay·xa=ay·xa {f+A (ay) ∩g + B (xa)} ⊇ f+A (ax) ∩g + B (ba) ∩f + A (ay) ∩g + B (xa) ⊇ f + A (a) ∩g + B (a), which shows that (f+A ∼ ◦g+B ) ∼ ◦(f+A ∼ ◦g+B ) ⊇ f + A∩g + B . Similarly we can show that (f − A ∼ ?g−B ) ∼ ?(f−A ∼ ?g−B ) ⊆ f − A∪g − B. Thus fA ugB v (fA �gB) � (fA �gB). Hence fA ugB = (fA �gB) � (fA �gB). Also by using Lemma 4.6, fA is DFS semiprime. (iv) =⇒ (iii) : Let R and L be any left and right ideals of S. Then by using Lemma 3.1, XR and XL are the DFS r-ideal and DFS l-ideal of S over U respectively. Now by using Lemma 3.4, we get XR∩L = XR uXL = (XR �XL) � (XR �XL) = (XR �XR) � (XL �XL) = X(R2] �X(L2] = X(R2L2] = X(L2R2], which implies that R∩L = (L2R2]. (iii) =⇒ (ii) : It is simple. (ii) =⇒ (i) : It can be followed from Corollary 4.3. � Theorem 4.8. Let S be an ordered A*G**-groupoid. Then the following conditions are equivalent: (i) S is strongly regular ; (ii) 〈R〉a2 ∩〈L〉a = (〈R〉a2 〈L〉a · 〈R〉a2 ] and 〈R〉a2 is semiprime; (iii) R∩L = (RL ·R] and R is semiprime; (iv) fA ugB = (fA �gB) �fA and fA is DFS semiprime; (v) S is strongly regular and |E|= 1, (a = ax ·a, ∀ a,x ∈ E); (vi) S is strongly regular and ∅ 6= E ⊆ S is semilattice. Proof. (i) =⇒ (vi) : It can be followed from Corollary 4.2 (i). (vi) =⇒ (v) : It can be followed from Corollary 4.2 (ii). (v) =⇒ (iv) : Let fA and gB be any DFS l-ideals of a strongly regular S over U. Now for a ∈ S, there exist some x,y ∈ S such that a ≤ ax ·ay ≤ ax ·(ax ·ay)y = ((ax ·ay)y ·x)a = (xy ·(ax ·ay))a = (ax ·(xy ·ay))a = (ax · (a · (xy)y))a. Int. J. Anal. Appl. 16 (4) (2018) 500 Thus (ax · (a · (xy)y),a) ∈ Aa. Therefore ((f+A ∼ ◦ g+B ) ∼ ◦ f+A )(a) = ⋃ (ax·(a·(xy)y),a)∈Aa {(f+A ∼ ◦ g+B )(ax · (a · (xy)y)) ∩g + B (a)} ⊇ ⋃ ax·(a·(xy)y≤ax·(a·(xy)y {f+A (ax) ∩g + B (a · (xy)y)}∩g + B (a) ⊇ f+A (ax) ∩g + B (a · (xy)y) ∩g + B (a) ⊇ f + A (a) ∩g + B (a), which shows that (f+A ∼ ◦g+B ) ∼ ◦f+A ⊇ f + A ∩g + B . Similarly we can show that (f − A ∼ ? g−B ) ∼ ? f−A ⊆ f − A ∪g − B. Thus (fA �gB) �fA w fA ugB. By using Lemmas 3.2 and 4.6, it is easy to show that (fA �gB) �fA v fA ugB. Thus fA ugB = (fA �gB) �fA. Also by using Lemma 4.6, fA is DFS semiprime. (iv) =⇒ (iii) : Let R and L be any left and right ideals of S. Then by Lemma 3.1, XR and XL are the DFS r-ideal and DFS l-ideal of S over U respectively. Now by using Lemmas 3.4, 4.1 and 2.1, we get XR∩L = XR uXL = (XR �XL) �XL = X((RL]·R] = X(RL·R], which shows that R∩L = (RL ·R]. Also by using Lemma 3.3, R is semiprime. (iii) =⇒ (ii) : It is simple. (ii) =⇒ (i) : Since (Sa2∪a2] and (Sa∪a] are the right and left ideals of S containing a2 and a respectively. Thus by using given assumption and Lemma 2.1, we get a ∈ (Sa2 ∪a2] ∩ (Sa∪a] = ((Sa2 ∪a2](Sa∪a] · (Sa2 ∪a2]] = ((Sa2 ∪a2)(Sa∪a) · (Sa2 ∪a2)] ⊆ (S(Sa∪a) · (Sa2 ∪a2)] = ((S2a∪Sa)(Sa2 ∪a2)] = ((S2a ·Sa2) ∪ (S2a ·a2) ∪ (Sa ·Sa2) ∪ (S2a ·a2)] ⊆ ((Sa ·a2S) ∪ (Sa ·Sa) ∪ (Sa ·a2S) ∪ (Sa ·Sa)] ⊆ ((Sa ·Sa) ∪ (Sa ·Sa) ∪ (Sa ·Sa) ∪ (Sa ·Sa)] = (Sa ·Sa] = (aS ·aS]. Hence S is strongly regular. � 5. Conclusions We have got some interesting and new characterizations which we usually do not find in other algebraic structures. We have considered the following problems in detail: i) Define and compare DFS left/right ideals of an ordered AG-groupoid and respective examples are provided. ii) Introduce the concept of an ordered A*G**-groupoid and characterize it by using DFS left/right ideals. Int. J. Anal. 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