International Journal of Analysis and Applications ISSN 2291-8639 Volume 14, Number 2 (2017), 193-202 http://www.etamaths.com INT-SOFT INTERIOR HYPERIDEALS OF ORDERED SEMIHYPERGROUPS ASGHAR KHAN1,∗, MUHAMMAD FAROOQ1 AND BIJAN DAVVAZ2 Abstract. The main theme of this paper is to study ordered semihypergroups in the context of int-soft interior hyperideals. In this paper, the notion of int-soft interior hyperideals are studied and their related properties are discussed. We present characterizations of interior hyperideals in terms of int-soft interior hyperideals. The concepts of int-soft hyperideals and int-soft interior hyperideals coincide in a regular as well as in intra-regular ordered semihypergroups. We prove that every int-soft hyperideal is an int-soft interior hyperideal but the converse is not true which is shown with help of an example. Furthermore we characterize simple ordered semihypergroups by means of int-soft hyperideals and int-soft interior hyperideals. 1. Introduction The real world is inherently uncertain, imprecise, and vague. Various problems in system identifi- cation involve characteristics which are essentially nonprobabilistic in nature [19]. In response to this situation, Zadeh [20], introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to un- certainty is outlined by Zadeh [21]. To solve a complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of intuitionistic fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [6]. Maji et al. [22] and Molodtsov [6], suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [6], introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Worldwide, there has been a rapid growth in interest in soft set theory and its applications in recent years see [1–5, 9, 10, 16]. The concept of hyperstructure was first introduced by Marty [7], at the 8th Congress of Scandinavian Mathematicians in 1934, when he defined hypergroups and started to analyze its properties. Now, the theory of algebraic hyperstruc- tures has become a well-established branch in algebraic theory and it has extensive applications in many branches of mathematics and applied science. Later on, people have developed the semihyper- groups, which are the simplest algebraic hyperstructures having closure and associative properties. A comprehensive review of the theory of hyperstructures can be found in [11–15, 17]. In this paper, we study the concept of int-soft interior hyperideals in ordered semihypergroups and present some related examples of this concept. We show that int-soft hyperideals and int-soft interior hyperideals coincide in regular ordered semihypergroups and intra-regular ordered semihypergroups. We characterize ordered semihypergroups in terms of int-soft hyperideals and int-soft interior hyperideals. Simple ordered Received 13th February, 2017; accepted 4th April, 2017; published 3rd July, 2017. 2010 Mathematics Subject Classification. 20N20. Key words and phrases. ordered semihypergroup; interior hyperideal; int-soft interior hyperideal; simple ordered semihypergroup; regular and intra-regular ordered semihypergroup . c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 193 194 KHAN, FAROOQ AND DAVVAZ semihypergroups are characterized by using the notions of int-soft hyperideals and int-soft interior hyperideals. 2. Preliminaries 2.1. Basic results on ordered semihypergroups. A hypergroupoid is a nonempty set S equipped with a hyperoperation ◦, that is a map ◦ : S × S −→ P∗ (S), where P∗ (S) denotes the set of all nonempty subsets of S (see [7]). We shall denote by x◦y, the hyperproduct of elements x,y of S. A hypergroupoid (S,◦) is called a semihypergroup if (x◦y) ◦z = x◦ (y ◦z) for all x,y,z ∈ S. Let A, B be the nonempty subsets of S. Then the hyperproduct of A and B is defined as A◦B = ⋃ a∈A,b∈B a◦b. We shall write A◦x instead of A◦{x} and x◦A for {x}◦A. Definition 2.1. (see [11]). An algebraic hyperstructure (S,◦,≤) is called an ordered semihypergroup (also called po-semihypergroup) if (S,◦) is a semihypergroup and (S,≤) is a partially ordered set such that the monotone condition holds as follows: a ≤ b implies that x◦a ≤ x◦ b and a◦x ≤ b◦x for all x,a,b ∈ S, where, if A,B ∈ P∗ (S) then we say that A � B if for every a ∈ A there exists b ∈ B such that a ≤ b. If A = {a} then we write a � B instead of {a}� B. Definition 2.2. (see [13]). A nonempty subset A of an ordered semihypergroup (S,◦,≤) is called a subsemihypergroup of S if for all x,y ∈ A, implies that x◦y ⊆ A. Equivalently A nonempty subset A of an ordered semihypergroup (S,◦,≤) is called a subsemihyper- group of S if A◦A ⊆ A. Definition 2.3. (see [11]). Let (S,◦,≤) be an ordered semihypergroup and A be a nonempty subset of S. Then A is called a left (resp., right) hyperideal of S if: (1) S ◦A ⊆ A (resp., A◦S ⊆ A). (2) If a ∈ A and S 3 b ≤ a then b ∈ A. If A is both a right hyperideal and a left hyperideal of S, then it is called a hyperideal (or two-sided hyperideal) of S. Definition 2.4. (see [18]) Let (S,◦,≤) be an ordered semihypergroup. A subsemihypergroup A of S is called an interior hyperideal of S if: (1) S ◦A◦S ⊆ A. (2) If a ∈ A and S 3 b ≤ a then b ∈ A. For A ⊆ S, we denote (A] = {t ∈ S | t ≤ h for some h ∈ A} . Lemma 2.1. (see [11]). Let (S,◦,≤) be an ordered semihypergroup and A,B are the nonempty subsets of S. Then the following statements hold: (1) A ⊆ (A] . (2) A ⊆ B implies that (A] ⊆ (B] . (3) (A] ◦ (B] ⊆ (A◦B] . (4) ((A] ◦ (B]] = (A◦B] . (5) ((A]] = (A] . Definition 2.5. (see [18]). An ordered semihypergroup (S,◦,≤) is called regular if for each a ∈ S there exists x ∈ S such that a ≤ a◦x◦a. Definition 2.6. (see [18]). An ordered semihypergroup (S,◦,≤) is called intra-regular if for each a ∈ S there exist x,y ∈ S such that a ≤ x◦a◦a◦y. 2.2. Basic concepts of soft sets. In what follows, we take E = S as the set of parameters, which is an ordered semihypergroup, unless otherwise specified. From now on, U is an initial universe set, E is a set of parameters, P(U) is the power set of U and A,B,C... ⊆ E. Definition 2.7. (see [6]). A soft set fA over U is defined as fA : E −→ P(U) such that fA(x) = ∅ if x /∈ A. Hence fA is also called an approximation function. A soft set fA over U can be represented by the set of ordered pairs fA = {(x,fA(x))|x ∈ E, fA(x) ∈ P(U)} . It is clear from Definition 2.7, that a soft set is a parameterized family of subsets of U. Note that the set of all soft sets over U will be denoted by S(U). Definition 2.8. (see [6]) (i) Let fA,fB ∈ S(U). Then fA is called a soft subset of fB, denoted by fA ⊆ fB if fA(x) ⊆ fB(x) for all x ∈ E. Two soft sets fA and fB are said to be equal soft sets if fA ⊆ fB and fB ⊆ fA and is INT-SOFT INTERIOR HYPERIDEALS OF ORDERED SEMIHYPERGROUPS 195 denoted by fA = fB. (ii) Let fA,fB ∈ S(U). Then the soft union of fA and fB, denoted by fA ∪fB = fA∪B, is defined by (fA ∪fB) (x) = fA(x) ∪fB(x) for all x ∈ E. (iii) Let fA,fB ∈ S(U). Then the soft intersection of fA and fB, denoted by fA∩fB = fA∩B, is defined by (fA ∩fB) (x) = fA(x) ∩fB(x) for all x ∈ E. For x ∈ S, we define Ax = {(y,z) ∈ S ×S | x ≤ y ◦z}. Definition 2.9. (see [8]). Let fA and gB be two soft sets of an ordered semihypergroup S over U. Then, the int-soft product, denoted by fA∗̃gB, is defined by fA∗̃gB : S −→ P(U),x 7−→ (fA∗̃gB) (x) = { ⋃ (y,z)∈Ax {fA(y) ∩gB(z)} , if Ax 6= ∅, ∅, if Ax = ∅, for all x ∈ S. Definition 2.10. (see [8]). For a nonempty subset A of S the characteristic soft set is defined to be the soft set SA of A over U in which SA is given as follows SA : S 7−→ P(U). x 7−→ { U, if x ∈ A ∅, otherwise For an ordered semihypergroup S, the soft set ”SS ” of S over U is defined as follows: SS : S −→ P(U),x 7−→SS(x) = U for all x ∈ S. The soft set ”SS” of an ordered semihypergroup S over U is called the whole soft set of S over U. Definition 2.11. (see [8]). Let fA be a soft set of an ordered semihypergroup S over U a subset δ such that δ ∈ P (U). The δ-inclusive set of fA is denoted by eA(fA,δ) and defined to be the set eA(fA,δ) = {x ∈ S | fA (x) ⊇ δ} . Definition 2.12. (see [8]). A soft set fA of an ordered semihypergroup S over U is called an int-soft subsemihypergroup of S over U if: (∀x,y ∈ S) ⋂ α∈x◦y fA(α) ⊇ fA(x) ∩fA(y). Definition 2.13. (see [8]). Let fA be a soft set of an ordered semihypergroup S over U. Then fA is called an int-soft left (resp., right) hyperideal of S over U if it satisfies the following conditions: (1) (∀x,y ∈ S) ⋂ α∈x◦y fA(α) ⊇ fA(y) ( resp., ⋂ α∈x◦y fA(α) ⊇ fA(x) ) . (2) (∀x,y ∈ S) x ≤ y =⇒ fA(x) ⊇ fA(y). A soft set fA over U is called an int-soft hyperideal (or int-soft two-sided hyperideal) of S over U if it is both an int-soft left hyperideal and an int-soft right hyperideal of S over U. Definition 2.14. An int-soft subsemihypergroup fA of an ordered semihypergroup S over U is called an int-soft interior hyperideal of S over U if it satisfies the following conditions: (1) (∀x,y,a ∈ S) ⋂ α∈x◦a◦y fA(α) ⊇ fA(a). (2) (∀x,y ∈ S) x ≤ y =⇒ fA(x) ⊇ fA(y). Example 2.1. Let (S,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a} {a} c {a} {a} {a} {a,b} d {a} {a} {a,b} {a,b,c} ≤:= {(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (b,d), (c,d)}. Suppose U = {p,q,r,s} and A = {a,b,c} . Let us define fA (a) = {p,q,r,s} , fA (b) = {p} , fA (c) = {p,q,r} and fA (d) = ∅. Then fA is an int-soft interior hyperideal of S over U. 196 KHAN, FAROOQ AND DAVVAZ Example 2.2. Let (S,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d e a {a,b} {a,b} {a,b} {a,b} {a,b} b {a,b} {a,b} {a,b} {a,b} {a,b} c {a,b} {a,b} {c} {c} {e} d {a,b} {a,b} {c} {d} {e} e {a,b} {a,b} {c} {c} {e} ≤:= {(a,a), (b,b), (c,c), (d,d), (e,e) , (a,c), (a,d), (a,e) , (b,c) , (b,d), (b,e) , (c,d), (c,e)}. Let U = {1, 2, 3, 4} and A = {a,b,c,e} . Let us define fA (a) = {1, 2, 3, 4} , fA (b) = {1, 2, 3, 4} , fA (c) = {3, 4} , fA (d) = ∅ and fA (e) = {3, 4} . Then fA is an int-soft interior hyperideal of S over U. Proposition 2.1. Let (S,◦,≤) be an ordered semihypergroup and A be a nonempty subset of S. Then A is an interior hyperideal of S if and only if the characteristic function SA is an int-soft interior hyperideal of S over U. Proof. Suppose that A is an interior hyperideal of S. Let x,y and a be any elements of S. If a ∈ A, then SA (a) = U. Since A is an interior hyperideal of S, we have α ∈ x◦a◦y ⊆ S◦A◦S ⊆ A we have SA (α) = U. Thus ⋂ α∈x◦a◦y SA (α) = U = SA (a) . If a /∈ A then SA (a) = ∅. Since SA (α) ⊇∅ = SA (a) . Thus ⋂ α∈x◦a◦y SA (α) ⊇SA (a) . Let x,y ∈ S with x ≤ y. If y /∈ A then SA (y) = ∅ and so SA (x) ⊇∅ = SA (y) . If y ∈ A then SA (y) = U. Since x ≤ y and A is an interior hyperideal of S, we have x ∈ A and thus SA (x) = U = SA (y) . Since A is an interior hyperideal of S. Therefore A is a subsemihypergroup of S. Let x,y ∈ S. Then we have, ⋂ α∈x◦y SA (α) ⊇SA (x)∩SA (y) for every α ∈ x◦y. Indeed: If x◦y * A, then there exists α ∈ x ◦ y such that α /∈ A, and we have ⋂ α∈x◦y SA (α) = ∅. Besides that x ◦ y * A implies that x /∈ A or y /∈ A. Then SA (x) = ∅ or SA (y) = ∅ and hence ⋂ α∈x◦y SA (α) = SA (x)∩SA (y) . Let x◦ y ⊆ A. Then SA (α) = U for any α ∈ x◦ y. It implies that ⋂ α∈x◦y SA (α) = U. Since we have SA (x) ⊆ U for any x ∈ A. Thus ⋂ α∈x◦y SA (α) ⊇ SA (x) ∩SA (y) . Therefore SA is an int-soft interior hyperideal of S over U. Conversely, let ∅ 6= A ⊆ S such that SA is an int-soft interior hyperideal of S over U. We claim that A◦A ⊆ A. To prove the claim, let x,y ∈ A. By hypothesis, ⋂ α∈x◦y SA (α) ⊇SA (x)∩SA (y) = U which implies that SA (α) ⊇ U for any α ∈ x◦y. On the other hand SA (x) ⊆ U for all x ∈ S. Thus for any α ∈ x◦y, SA (α) = U implies that α ∈ A. It thus follows that A◦A ⊆ A. Let α ∈ S◦A◦S, then there exist x,y ∈ S and a ∈ A such that α ∈ x◦a◦y. Since ⋂ α∈x◦a◦y SA (α) ⊇ SA (a) , and a ∈ A we have SA (a) = U. Hence for each α ∈ S◦A◦S, we have SA (α) = U, and so α ∈ A. Thus S◦A◦S ⊆ A. Let x ∈ S and y ∈ A be such that x ≤ y. Then SA (x) ⊇ SA (y) = U, and thus x ∈ A. Therefore A is an interior hyperideal of S. Proposition 2.2. Let (S,◦,≤) be an ordered semihypergroup and fA be an int-soft hyperideal of S over U. Then fA is an int-soft interior hyperideal of S over U. Proof. Let x,a,y ∈ S. Since fA is an int-soft hyperideal of S over U. Then for any α ∈ x◦a◦y, we have ⋂ α∈x◦a◦y fA (α) = ⋂ α∈x◦β β∈a◦y fA (α) ⊇ fA (β) ⊇ ⋂ β∈a◦y fA (β) ⊇ fA (a) . Thus ⋂ α∈x◦a◦y fA (α) ⊇ fA (a) . Therefore fA is an int-soft interior hyperideal of S over U. The converse of Proposition 2.2, is not true in general. We can illustrate it by the following example. Example 2.3. Let (S,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a} {a} c {a} {a} {a,b} {a,b} d {a} {a} {a,b} {a} INT-SOFT INTERIOR HYPERIDEALS OF ORDERED SEMIHYPERGROUPS 197 ≤:= {(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (d,b), (d,c)}. Suppose U = {p,q,r} and A = {a,b,d} . Let us define fA (a) = {p,q,r} , fA (b) = {p} , fA (c) = ∅ and fA (d) = {p,r} . Then fA is an int-soft interior hyperideal of S over U. This is not an int-soft left hyperideal as ⋂ α∈c◦d={a,b} fA (a) ∩fA (b) = {p} + {p,r} = fA (d) . Proposition 2.3. Let (S,◦,≤) be a regular ordered semihypergroup and fA is an int-soft interior hyperideal of S over U. Then fA is an int-soft hyperideal of S over U. Proof. Let x,y ∈ S. Since fA is an int-soft interior hyperideal of S over U. Then ⋂ α∈x◦y fA (α) ⊇ fA (x) . Indeed: Since S is regular and x ∈ S, then there exists z ∈ S such that x ≤ x ◦ z ◦ x. Then we have x ◦ y ≤ (x◦z ◦x) ◦ y = (x◦z) ◦ (x◦y) . So there exist α ∈ x ◦ y, v ∈ x ◦ z and β ∈ v ◦ x ◦ y such that α ≤ β. Since fA is an int-soft interior hyperideal of S over U, we have fA (α) ⊇ fA (β) ⊇ ⋂ β∈v◦x◦y fA (β) ⊇ fA (x) . Thus ⋂ α∈x◦y fA (α) ⊇ fA (x) . Therefore fA is an int-soft right hyperideal of S over U. In a similar way we prove that fA is an int-soft left hyperideal of S over U. By Propositions 2.2 and 2.3 we have the following: Theorem 2.1. In regular ordered semihypergroups the concepts of int-soft hyperideals and int-soft interior hyperideals coincide. Proposition 2.4. Let (S,◦,≤) be an intra-regular ordered semihypergroup and fA is an int-soft interior hyperideal of S over U. Then fA is an int-soft hyperideal of S over U. Proof. Let a,b ∈ S. Then ⋂ u∈a◦b fA (u) ⊇ fA (a) . Indeed: Since S is intra-regular and a ∈ S, there exist x,y ∈ S such that a ≤ x◦a◦a◦y. Then a◦ b ≤ (x◦a◦a◦y) ◦ b = x◦a◦ (a◦y ◦ b). So there exist u ∈ a◦b, v ∈ a◦y◦b and α ∈ x◦a◦v such that u ≤ α. Since fA is an int-soft interior hyperideal of S over U, we have fA (u) ⊇ fA (α) ⊇ ⋂ α∈x◦a◦v fA (α) ⊇ fA (a) . Thus ⋂ u∈a◦b fA (u) ⊇ fA (a) . Hence fA is an int-soft right hyperideal of S over U. Similarly we can prove that fA is an int-soft left hyperideal of S over U. Therefore fA is an int-soft hyperideal of S over U. By Propositions 2.2 and 2.4 we have the following: Theorem 2.2. In intra-regular ordered semihypergroups the concepts of int-soft hyperideals and int- soft interior hyperideals coincide. Theorem 2.3. Let fA be a soft set of an ordered semihypergroup S over U and δ ∈ P (U) . Then fA is an int-soft interior hyperideal of S over U if and only if each nonempty δ-inclusive set eA(fA,δ) is an interior hyperideal of S. Proof. Assume that fA is an int-soft interior hyperideal of S over U. Let δ ∈ P (U) such that eA(fA,δ) 6= ∅. Let x,y ∈ eA(fA,δ). Then fA (x) ⊇ δ and fA (y) ⊇ δ. By hypothesis, we have⋂ α∈x◦y fA (α) ⊇ fA (x) ∩ fA (y) ⊇ δ ∩ δ = δ. Thus for any α ∈ x ◦ y, we have fA (α) ⊇ δ, implies that α ∈ eA(fA,δ). It follows that x◦y ⊆ eA(fA,δ). Hence eA(fA,δ) is a subsemihypergroup of S. Let y ∈ eA(fA,δ) and x,z ∈ S. Then fA (y) ⊇ δ. Since fA is an int-soft interior hyperideal of S over U. Thus ⋂ w∈x◦y◦z fA (w) ⊇ fA (y) ⊇ δ. Hence fA (w) ⊇ δ for any w ∈ x◦y ◦z implies that w ∈ eA(fA,δ). Thus S◦eA(fA,δ)◦S ⊆ eA(fA,δ). Let x ∈ eA(fA,δ) and y ∈ S with y ≤ x. Then δ ⊆ fA (x) ⊆ fA (y) , we get y ∈ eA(fA,δ). Therefore eA(fA,δ) is an interior hyperideal of S. Conversely, suppose that eA(fA,δ) 6= ∅ is an interior hyperideal of S. If ⋂ α∈x◦y fA (α) ⊂ fA (x)∩fA (y) for some x,y ∈ S, then there exists δ ∈ P (U) such that ⋂ α∈x◦y fA (α) ⊂ δ ⊆ fA (x) ∩ fA (y) , which implies that x,y ∈ eA(fA,δ) and x◦y * eA(fA,δ). It contradicts the fact that eA(fA,δ) is an interior hyperideal of S. Consequently, ⋂ α∈x◦y fA (α) ⊇ fA (x) ∩ fA (y) for all x,y ∈ S. Next we show that⋂ α∈x◦a◦y fA (α) ⊇ fA (a) for all x,a,y ∈ S. Choose fA (a) = δ, then a ∈ eA(fA,δ). Since eA(fA,δ) is an interior hyperideal of S, we get x◦a◦y ⊆ eA(fA,δ). Then for every α ∈ x◦a◦y, we have fA (α) ⊇ δ and so fA (a) = δ ⊆ ⋂ α∈x◦a◦y fA (α) . Let x,y ∈ S such that x ≤ y. lf fA(y) = δ then y ∈ eA(fA,δ). Since eA(fA,δ) is an interior hyperideal of S, we get x ∈ eA(fA,δ). So fA(x) ⊇ δ = fA(y). Therefore fA is an int-soft interior hyperideal of S over U. 198 KHAN, FAROOQ AND DAVVAZ Example 2.4. Let (S,◦,≤) be an ordered semihypergroup where the hyperoperation and the order relation are defined by: ◦ a b c d a {a} {a} {a} {a} b {a} {a} {a,d} {a} c {a} {a} {a} {a} d {a} {a} {a} {a} ≤:= {(a,a), (b,b), (c,c), (d,d), (a,d)}. Then the interior hyperideals of S are {a} , {a,b} , {a,c} , {a,d} , {a,b,d} , {a,c,d} and S. Suppose U = {e1,e2,e3,e4} and A = {a,b,d} . Let us define fA (a) = {e1,e2,e3,e4} , fA (b) = {e1,e3} , fA (c) = ∅ and fA (d) = {e1,e4} . Then eA(fA,δ) =   {a,b,d} if δ = {e1} {a} if δ = {e2} {a,b} if δ = {e3} {a,d} if δ = {e4} {a} if δ = {e1,e2} {a,b} if δ = {e1,e3} {a,d} if δ = {e1,e4} {a} if δ = {e2,e3} {a} if δ = {e2,e4} {a} if δ = {e3,e4} {a} if δ = {e1,e2,e3} {a} if δ = {e1,e2,e4} {a} if δ = {e1,e3,e4} {a} if δ = {e2,e3,e4} {a} if δ = U So by Theorem 2.3, fA is an int-soft interior hyperideal of S over U. Theorem 2.4. Let {fAi | i ∈ I} be a family of int-soft interior hyperideals of an ordered semi- hypergroup S over U. Then fA = ⋂ i∈I fAi is an int-soft interior hyperideal of S over U where(⋂ i∈I fAi ) (x) = ⋂ i∈I (fAi (x)) . Proof. Let x,y ∈ S. Then, since each fAi (i ∈ I) is an int-soft interior hyperideals of S over U, so ⋂ α∈x◦y fAi (α) ⊇ fAi (x) ∩fAi (y) . Thus for any α ∈ x◦y, fAi (α) ⊇ fAi (x) ∩fAi (y) , and we have fA (α) = (⋂ i∈I fAi ) (α) = ⋂ i∈I (fAi (α)) ⊇ ⋂ i∈I (fAi (x) ∩fAi (y)) = (⋂ i∈I (fAi (x)) ) ∩ (⋂ i∈I (fAi (y)) ) =(⋂ i∈I fAi ) (x) ∩ (⋂ i∈I fAi ) (y) = fA (x) ∩ fA (y) , which implies that ⋂ α∈x◦y fA (α) ⊇ fA (x) ∩ fA (y) . Let a,x,y ∈ S and ⋂ β∈x◦a◦y fAi (β) ⊇ fAi (a) . Thus for any β ∈ x ◦ a ◦ y, fAi (β) ⊇ fAi (a) . Then fA (β) = (⋂ i∈I fAi ) (β) = ⋂ i∈I (fAi (β)) ⊇ ⋂ i∈I (fAi (a)) = (⋂ i∈I fAi ) (a) = fA (a) . Thus⋂ β∈x◦a◦y fA (β) ⊇ fA (a). Furthermore, if x ≤ y, then fA (x) ⊇ fA (y) . Indeed: Since every fAi (i ∈ I) is an int-soft interior hyperideal of S over U, it can be obtained that fAi (x) ⊇ fAi (y) for all i ∈ I. Thus fA (x) = (⋂ i∈I fAi ) (x) = ⋂ i∈I (fAi (x)) ⊇ ⋂ i∈I (fAi (y)) = (⋂ i∈I fAi ) (y) = fA (y) . Thus fA is an int-soft interior hyperideals of S over U. Lemma 2.2. Let S be an ordered semihypergroup and fA is a soft set of S over U. If fA is an int-soft subsemihypergroup of S over U such that x ≤ y =⇒ fA(x) ⊇ fA(y), ∀x,y ∈ S, INT-SOFT INTERIOR HYPERIDEALS OF ORDERED SEMIHYPERGROUPS 199 then fA∗̃fA⊆̃fA. Conversely if fA∗̃fA⊆̃fA, then fA is an int-soft subsemihypergroup of S over U. Proof. Let x ∈ S. If Ax = ∅, then (fA∗̃fA) (x) = ∅ ⊆ fA (x) . If Ax 6= ∅, then (b,c) ∈ Ax such that x ≤ b◦ c. This means that there exists α ∈ b◦ c such that x ≤ α. (fA∗̃fA) (x) = ⋃ (b,c)∈Ax {fA (b) ∩fA (c)} ⊆ ⋃ (b,c)∈Ax fA (α) ⊆ ⋃ (b,c)∈Ax fA (x) = fA (x) . Thus fA∗̃fA ⊆ fA. Conversely, if fA∗̃fA ⊆ fA, then for all x,y ∈ S and α ∈ x◦y. We have fA (α) ⊇ (fA∗̃fA) (α) = ⋃ (x,y)∈Aα {fA (x) ∩fA (y)} ⊇ {fA (x) ∩fA (y)} fA (α) ⊇ {fA (x) ∩fA (y)} . Hence ⋂ α∈x◦y fA (α) ⊇{fA (x) ∩fA (y)} . Thus fA is an int-soft subsemihypergroup of S over U. Theorem 2.5. Let (S,◦,≤) be an ordered semihypergroup and fA be a soft set of S over U. Then fA is an int-soft interior hyperideal of S over U if and only if fA∗̃fA ⊆ fA and SS∗̃fA∗̃SS ⊆ fA. Proof. Let fA be an int-soft interior hyperideal of S over U. Then (SS∗̃fA∗̃SS) (a) ⊆ fA (a) for all a ∈ S. Indeed: If (SS∗̃fA∗̃SS) (a) = ∅, clearly, (SS∗̃fA∗̃SS) (a) ⊆ fA (a) . Let (SS∗̃fA∗̃SS) (a) 6= ∅. Then we can prove that (SS∗̃fA∗̃SS) (a) ⊆ fA (a) . In fact, let (x,y) ∈ Aa and (p,q) ∈ Ax, i.e., a ≤ x◦y and x ≤ p◦q. Then a ≤ p◦q◦y, and there exists u ∈ p◦q◦y such that a ≤ u. Since fA is an int-soft interior hyperideal of S over U. Then fA (a) ⊇ fA (u) ⊇ ⋂ u∈p◦q◦y fA (u) ⊇ fA (q) . Thus (SS∗̃fA∗̃SS) (a) = ⋃ (x,y)∈Aa {(SS∗̃fA) (x) ∩SS (y)} = ⋃ (x,y)∈Aa {(SS∗̃fA) (x) ∩U} = ⋃ (x,y)∈Aa (SS∗̃fA) (x) = ⋃ (x,y)∈Aa   ⋃ (p,q)∈Ax (SS (p) ∩fA (q))   = ⋃ (x,y)∈Aa   ⋃ (p,q)∈Ax (U ∩fA (q))   = ⋃ (x,y)∈Aa ⋃ (p,q)∈Ax (fA (q)) ⊆ fA (a) . Thus SS∗̃fA∗̃SS ⊆ fA. Conversely, for any x,y,z ∈ S, let α ∈ x ◦ y ◦ z. Then, there exists u ∈ x ◦ y ⊆ (x◦y] such that 200 KHAN, FAROOQ AND DAVVAZ α ∈ u◦z ⊆ (u◦z] , and we have (x,y) ∈ Au, (u,z) ∈ Aα. Since SS∗̃fA∗̃SS ⊆ fA, we have fA (α) ⊇ (SS∗̃fA∗̃SS) (α) = ⋃ (p,q)∈Aα [{SS∗̃fA}(p) ∩SS (q)] ⊇ {(SS∗̃fA) (u) ∩SS (z)} = {(SS∗̃fA) (u) ∩U} = (SS∗̃fA) (u) = ⋃ (s,t)∈Au [SS (s) ∩fA (t)] ⊇ {SS (x) ∩fA (y)} = {U ∩fA (y)} = fA (y) . It thus follows that ⋂ α∈x◦y◦z fA (α) ⊇ fA (y) . The rest of the proof is a consequence of the Lemma 2.2. 3. Characterizations of simple ordered semihypergroups in terms of int-soft hyperideals and int-soft interior hyperideals Definition 3.1. (see [18]). An ordered semihypergroup (S,◦,≤) is called simple if it has no a proper hyperideal. Lemma 3.1. (see [18]). An ordered semihypergroup (S,◦,≤) is a simple ordered semihypergroup if and only if for every a ∈ S, (S ◦a◦S] = S. Let (S,◦,≤) is an ordered semihypergroup and a ∈ S, and fA be a soft set of S over U we denote by Ia the subset of S defines as follows: Ia = {b ∈ S | fA (b) ⊇ fA (a)} . Proposition 3.1. Let (S,◦,≤) be an ordered semihypergroup and fA is an int-soft right hyperideals of S over U. Then the set Ia is a right hyperideal of S for every a ∈ S. Proof. Let a ∈ S. First of all ∅ 6= Ia ⊆ S. Since a ∈ Ia. Let b ∈ Ia and s ∈ S. Then b◦s ⊆ Ia. Indeed: Since fA is an int-soft right hyperideals of S over U and b,s ∈ S, we have ⋂ α∈b◦s fA (α) ⊇ fA (b) . Since b ∈ Ia, we have fA (b) ⊇ fA (a) . Thus ⋂ α∈b◦s fA (α) ⊇ fA (a) , implies that fA (α) ⊇ fA (a) , so α ∈ Ia and hence b◦ s ⊆ Ia. Let b ∈ Ia and S 3 s ≤ b. Then s ∈ Ia. Indeed: Since fA is an int-soft right hyperideals of S over U, b,s ∈ S and s ≤ b, we have fA (s) ⊇ fA (b) . Since b ∈ Ia, we have fA (b) ⊇ fA (a) . Then fA (s) ⊇ fA (a) , so s ∈ Ia. In a similar way we prove the following: Proposition 3.2. Let (S,◦,≤) be an ordered semihypergroup and fA is an int-soft left hyperideals of S over U. Then the set Ia is a left hyperideal of S for every a ∈ S. By Propositions 3.1 and 3.2 we have the following: Proposition 3.3. Let (S,◦,≤) be an ordered semihypergroup and fA is an int-soft hyperideals of S over U. Then the set Ia is a hyperideal of S for every a ∈ S. Theorem 3.1. (see [8]). Let (S,◦,≤) be an ordered semihypergroup and ∅ 6= I ⊆ S. Then I is a hyperideal of S if and only if the characteristic function SI is an int-soft hyperideals of S over U. Theorem 3.2. An ordered semihypergroup (S,◦,≤) is a simple ordered semihypergroup if and only if every int-soft hyperideal of S over U is a constant function. Proof. Assume that S is a simple ordered semihypergroup. Let fA is an int-soft hyperideal of S over U and a,b ∈ S. By Proposition 3.3, we obtain Ia is a hyperideal of S. By assumption, this implies that Ia = S. Then b ∈ Ia, that is fA (b) ⊇ fA (a) . By symmetry we get fA (a) ⊇ fA (b) . Therefore fA (a) = fA (b) . Conversely, we assume that for every int-soft hyperideal of S over U is a constant function. Let I be a hyperideal of S and x ∈ S. By Theorem 3.1, we obtain the characteristic function SI is an int-soft hyperideal of S over U. By assumption, SI is a constant function, that is SI (x) = SI (b) for every INT-SOFT INTERIOR HYPERIDEALS OF ORDERED SEMIHYPERGROUPS 201 b ∈ S. Let a ∈ I. Then SI (x) = SI (a) = U, and so x ∈ I. Therefore S ⊆ I. Theorem 3.3. Let (S,◦,≤) be an ordered semihypergroup. Then S is a simple ordered semihyper- group if and only if every int-soft interior hyperideal of S over U is a constant function. Proof. Assume that S is a simple ordered semihypergroup. Let fA be an int-soft interior hyperideal of S over U and a,b ∈ S. By Lemma 3.1, we have S = (S ◦ b◦S] . Since a ∈ S, we have a ∈ (S ◦ b◦S] . Then there exist x,y ∈ S such that a ≤ x◦ b◦y, i.e., there exists α ∈ x◦ b◦y such that a ≤ α. Since fA is an int-soft interior hyperideal of S over U, we have fA (a) ⊇ fA (α) ⊇ ⋂ α∈x◦b◦y fA (α) ⊇ fA (b) . Hence fA (a) ⊇ fA (b) . By symmetry we can prove that fA (b) ⊇ fA (a) . Therefore fA (a) = fA (b) . Conversely, assume that every int-soft interior hyperideal of S over U is a constant function. Let fA is an int-soft hyperideal of S over U. Then fA is an int-soft interior hyperideal of S over U. By assumption fA is a constant function. By Theorem 3.2, S is a simple ordered semihypergroup. Corollary 3.1. Let (S,◦,≤) be an intra-regular ordered semihypergroup. Then every int-soft interior hyperideal of S over U is a constant function. As a consequence of Lemma 3.1, Theorem 3.2, and Theorem 3.3, we present characterizations of a simple ordered semihypergroup as the following theorem. Theorem 3.4. Let (S,◦,≤) be an ordered semihypergroup. Then the following statements are equiv- alent: (1) S is a simple ordered semihypergroup. (2) S = (S ◦a◦S] for every a ∈ S. (3) Every int-soft hyperideal of S over U is a constant function. (4) Every int-soft interior hyperideal of S over U is a constant function. Proposition 3.4. Let (S,◦,≤) be an intra-regular ordered semihypergroup. Then for every interior hyperideals A and B of S we have (1) (A◦A] = A. (2) (A◦B] = (B ◦A] . Proof. (1) . Let S be an intra-regular ordered semihypergroup and A, B are the interior hyperide- als of S. Let a ∈ A. Since S is intra-regular, there exist x,y ∈ S such that a ≤ x ◦ a ◦ a ◦ y = (x◦a)◦(a◦y) ≤ x◦(x◦a◦a◦y)◦(x◦a◦a◦y)◦y = ((x◦x◦a) ◦a◦ (y))◦((x◦a) ◦a◦ (a◦y ◦y)) ⊆ (S ◦A◦S) ◦ (S ◦A◦S) ⊆ A◦A =⇒ a ∈ (A◦A] =⇒ A ⊆ (A◦A] . For the reverse inclusion, let a ∈ (A◦A] , then a ≤ a1 ◦ a2 for some a1,a2 ∈ A. Then a ≤ x◦a◦a◦y = (x◦a)◦(a◦y) ≤ x◦(a1 ◦a2)◦(a1 ◦a2)◦y = (x◦a1 ◦a2)◦a1 ◦(a2 ◦y) ⊆ S◦A◦S ⊆ A =⇒ a ∈ (A] = A =⇒ (A◦A] ⊆ A. Thus (A◦A] = A. (2) . Let A and B be interior hyperideals of S. Then (A◦B] = (B ◦A] . Indeed: By (1) we have (A◦B] = ((A◦B] ◦ (A◦B]] = (((A◦B] ◦ (A◦B]) ◦ ((A◦B] ◦ (A◦B])] ⊆ (((A◦B) ◦ (A◦B)] ◦ ((A◦B) ◦ (A◦B)]] = (((A) ◦B ◦ (A◦B)] ◦ ((A◦B) ◦A◦ (B)]] ⊆ ((S ◦B ◦S] ◦ (S ◦A◦S]] ⊆ ((B] ◦ (A]] = (B ◦A] =⇒ (A◦B] ⊆ (B ◦A] . By symmetry we have (B ◦A] ⊆ (A◦B] . Thus (A◦B] = (B ◦A] . Proposition 3.5. Let (S,◦,≤) be an intra-regular ordered semihypergroup and fA is an int-soft interior hyperideal of S over U. Then for every a ∈ S such that a◦a ≤ a, we have the following (1) ⋂ v∈a◦a fA (v) = fA (a) . (2) ⋂ α∈a◦b fA (α) = ⋂ β∈b◦a fA (β) . Proof. (1) . Let S be an intra-regular ordered semihypergroup and fA is an int-soft interior hyperideal of S over U and a ∈ S. Then ⋂ v∈a◦a fA (v) = fA (a) . Indeed: Since S is intra-regular and a ∈ S, there exist x,y ∈ S such that a ≤ x◦a◦a◦y for some x,y ∈ S. So there exist v ∈ a◦a and z ∈ x◦v ◦y such that a ≤ z. Then fA (a) ⊇ fA (z) ⊇ ⋂ z∈x◦v◦y fA (z) ⊇ fA (v) . Hence fA (a) ⊇ ⋂ v∈a◦a fA (v) . Since a◦a ≤ a so there is v ∈ a◦a such that v ≤ a. 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