Abstract
FUZZY INFORMATION AND ENGINEERING 2019, VOL. 11, NO. 2, 127–148 https://doi.org/10.1080/16168658.2021.1886816 Semihypergroups Characterized by Generalized Fuzzy Bi-Hyperideals a b b Saleem Abdullah , Kostaq Hila and Shkelqim Kuka a b Department of Mathematics, Hazara University, Mansehra, Pakistan; Department of Mathematical Engineering, Faculty of Mathematics Engineering and Physics Engineering, Polytechnic University of Tirana, Tirana, Albania ABSTRACT ARTICLE HISTORY Received 22 October 2018 In this paper, we introduce the notion of interval valued (α, β)-fuzzy Revised 25 November 2018 bi-hyperideal in semihypergroups. The obtained concept is a gen- Accepted 16 March 2019 eralized form of fuzzy bi- hyperideal and (α, β)-fuzzy bi-hyperideal in semihypergroups, where α, β ∈{∈, q, ∈∨q, ∈∧q} with α =∈ ∧q. KEYWORDS So, we can easily construct twelve different types of interval val- Semihypergroups; interval ued fuzzy left (right) hyperideals of semihypergroups. Combining the valued (αβ)-fuzzy notion of an interval valued fuzzy point and an interval valued fuzzy bi-hyperideal; interval valued (∈∈ ∨q)-fuzzy set, we introduce the notion of a generalized interval valued fuzzy bi-hyperideals bi- hyperideal in semihypergroups and some useful characterization theorems are provided. We give some special attention to interval 2000 MATHEMATICS valued (∈, ∈∨q)-fuzzy bi-hyperideals. SUBJECT CLASSIFICATIONS 20N20; 08A72; 20N25 1. Introduction The basic concept of fuzzy set initiated by Zadeh [1] in 1965. After the semblance of fuzzy set theory, Rosenfeld studied fuzzy subgroup of a group [2]. Bhakat and Das [3,4] generalized the concept of Rosenfeld fuzzy group by using the idea of ‘belongingness’ and ‘quasico- incidence’ to define (∈, ∈∨q)-fuzzy subgroups. Some basic results on fuzzy ideals can be found in [4–10]. The algebraic hyperstructures represent a natural generalization of classical algebraic structures which is based on the notion of hyperoperation introduced by the French math- ematician Marty [11] in 1934. After Marty’s work, in the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many math- ematicians. A lot of papers and several books have been written on hyperstructure theory. A recent book on hyperstructures [12] shows great applications of algebraic hyperstructures in fuzzy set theory, automata, hypergraphs, binary relations, lattices, and probabilities. The relationships of hyperstructures and fuzzy sets have considered by Davvaz [13–17], Corsini [18–21], Leoreanu [22], Tofan [23], Kehagias [24], Cristea et al. [25,26] and others. In [27], by combining the notion of a fuzzy point and a fuzzy set, the authors introduced the notion of CONTACT Kostaq Hila kostaq_hila@yahoo.com © 2021 The Author(s). Published by Taylor & Francis Group on behalf of the Fuzzy Information and Engineering Branch of the Operations Research Society, Guangdong Province Operations Research Society of China. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 128 S. ABDULLAH ET AL. (α, β)-fuzzy hyperideals and (∈, ∈∨q)-fuzzy hyperideals in semihypergroups. This is a gen- eralization of the concept of a fuzzy right (resp. left) hyperideal of a semihypergroup and some useful theorems are obtained. In [28], Jun et. al. initiated the notion of an (α, β)-fuzzy bi-ideal in ordered semigroups and some useful theorems are studied (cf. [29–31]). See also [32–35]. In this paper, we worked on interval valued (α, β)-fuzzy bi-hyperideals of semihyper- groups and obatained some usful results. The obtained concept is a generalized form of fuzzy hyperideals and (α, β)-fuzzy bi-hyperideals in semihypergroups, where α, β ∈{∈, q, ∈ ∨q, ∈∧q} with α =∈ ∧q. Several characterization theorems are provided. 2. Definitions and Basic Results ∗ ∗ A hyperoperation is a map ◦ : S × S → P (S), where P (S) denotes the power set of S [12]. We shall denote by x ◦ y, the hyperoperation 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. Throughout this paper S will denote a semihypergroup with hyperoperation °. Let P, Q be two non-empty subsets of S. Then, the hyperproduct of P and Q is defined as: P ◦ Q = p ◦ q. p∈P,q∈Q We shall write A ◦ x instead of A ◦{x} and x ◦ A for {x}◦ A. A non-empty subset H of a semihypergroup S is called a subsemihypergroup of S if for all x, y ∈ H, x ◦ y ⊆ H. If a semihypergroup S contains an element e with the property that, for all x ∈ S, x ∈ x ◦ e (resp. x ∈ e ◦ x), we say that e is a right (resp. left) identity of S.If x ◦ e ={x} (resp. e ◦ x = {x}), for all x in S, then e is called scalar right (resp. left) identity in S. In [36], it is defined that if A ∈ P (S), then A is called, (i) a right hyperideal in S if x ∈ A ⇒ x ◦ y ⊆ A; ∀ y ∈ S. (ii) a left hyperideal in S if x ∈ A ⇒ y ◦ x ⊆ A; ∀ y ∈ S. (iii) a hyperideal in S if it is both a left and a right hyperideal in S. A fuzzy subset A in a set S of the form t =0if y = x A(y) = 0if y = x is said to be a fuzzy point with support x and value t and is denoted x . A fuzzy point x is t t said to be belong to (resp. be quasi-coincident with) a fuzzy set A, written as x ∈ A (resp. x qA)if A(x) ≥ t(resp. A(x) + t > 1). If x ∈ A or x qA, then we write x ∈ qA. The symbol∈∨q t t t t means neither ∈ nor q hold. FUZZY INFORMATION AND ENGINEERING 129 An interval number [37] on [0, 1], denoted by a, is defined as the closed subinterval of − + − + [0, 1], where a = [a , a ] satisfying 0 ≤ a ≤ a ≤ 1. We denote D[0, 1] as the set of all interval numbers on [0, 1] and also denote the interval numbers [0, 0] and [1, 1] by 0and 1 respectively. The interval [a, a] can be simply identified by the number a ∈ [0, 1]. − + − + We define the following for the interval number a = [a , a ], b = [b , b ] for all i ∈ I : i i i i i i − − + + (i) r max{ a , b }= [max(a , b ),max(a , b )], i i i i i i − − + + (ii) r min{ a , b }= [min(a , b ),min(a , b )], i i i i i i − + − + (iii) r inf a = [ a , a ], r sup a = [ a , a ] i i i∈I i i∈I i i∈I i i∈I i − − + + (iv) a ≤ a ⇔ a ≤ a and a ≤ a 1 2 1 2 1 2 − − + + (v) a = a ⇔ a = a and a = a 1 2 1 2 1 2 − − + + (vi) a < a ⇔ a < a and a < a 1 2 1 2 1 2 − + (vii) k a = [ka , ka ] 1 1 Let a and b be two interval numbers. The arithmetic operation +, −, ·, / may be extended to pairs of interval numbers as follows: − − + + a + b = [a + b , a + b ] − + + − a − b = [a − b , a − b ] − − + + a · b = [a · b , a · b ] − + + − − + a/b = [a , a ] · [1/b ,1/b ] for 0 ∈ / [b , b ]. Note:Wewrite a ≥ b whenever b ≤ a and a > b whenever b < a. In this paper we assume that any two interval numbers in D[0, 1] are comparable i.e. for any two interval numbers a and b in D[0, 1], we have either a ≤ b or a > b. It is clear that (D[0, 1], ≤, ∨, ∧) is a complete lattice with 0 = [0, 0] as the least ele- ment and 1 = [1, 1] as the greatest element. By an interval valued fuzzy set μ on X,we − + − + mean the set, μ ={(x,[μ (x), μ (x)])|x ∈ X}, where μ and μ are two fuzzy subsets μ μ − + − + of X such that μ (x) ≤ μ (x) for all x ∈ X. Putting μ = [μ (x), μ (x)], then we see that μ ={(x, μ(x))|x ∈ X}, where μ : X → D[0, 1]. Let S be a semihypergroup. By an interval valued fuzzy subsetA of S, we mean a mapping A : S → D[0, 1]. For any interval valued fuzzy subset A of S and for any t ∈ D[0, 1], U(A; t) ={x ∈ S : A(x) ≥ t} is called a t-level subset of A. For any two interval valued fuzzy subsets A and B of S, A ≤ B means that, for all x ∈ S, A(x) ≤ B(x). For x ∈ S, define X ={(y, z) ∈ S × S : x ∈ y ◦ z}. For any two interval valued fuzzy susbets A and B of S, define min{A(y), B(z)} if X =∅ A ◦ B : S → D[0, 1]|x −→ (y,z)∈X 0if X =∅ x 130 S. ABDULLAH ET AL. For a non-empty family of interval valued fuzzy subsets, A and A of S are defined i i i∈I i∈I as follows: A : S −→ A (x) := sup{A (x)} i i i i∈I i∈I i∈I and A : S −→ A (x) := inf{A (x)}. i i i i∈I i∈I i∈I If I is a finite set, say I ={1, 2, 3, ... , n}, then clearly A (x) := max{A (x), A (x), ... , A (x)}. i 1 2 n i∈I and A (x) := min{A (x), A (x), ... , A (x)}. i 1 2 n i∈I If A ⊆ S, then the interval valued characteristic function C of A is the interval valued fuzzy set in S, defined as follows: 1if x ∈ A C : S → D[0, 1]|x −→ C(x) = 0if x ∈ / A The following results are straightforward. Proposition 2.1: If A, B are subsets of a set X, then A ⊆ B if and only if A ≤ A . A B Corollary 2.2: Let A, B be subsets of a set X, then A = B if and only if A = A . A B Proposition 2.3: Let A, B be susbets of a set X, then A = A ∧ A . A∩B A B Proposition 2.4: Let (S, ◦) be a semihypergroup and A, B be subsets of S. Then A ◦ A = A B A . A◦B Definition 2.5: Let S be a semihypergroup and B be an interval valued fuzzy set in S. Then, B is called an interval valued fuzzy right hyperideal of S if B(x) ≤ inf {B(α)}, for every α∈x◦y x, y ∈ S; B is called an interval valued fuzzy left hyperideal of S if B(y) ≤ inf {B(α)}, for every α∈x◦y x, y ∈ S; B is called an interval valued fuzzy hyperideal of S if B is an interval valued fuzzy left hyperideal and an interval valued fuzzy right hyperideal of S. The following results are straightforward. FUZZY INFORMATION AND ENGINEERING 131 Lemma 2.6: Let B be an interval valued fuzzy hyperideal of S. Then max{B(x ), ... , B(x )}≤ inf {B(α)}, for all x , ... , x ∈ . 1 n 1 n α∈x ◦x ◦··· .◦x 1 2 n Proposition 2.7: A non-empty subset A of S is a hyperideal of S if and only if the interval valued characteristic function A of A is an interval valued fuzzy hyperideal of S. Proposition 2.8: LetA be an an interval valued fuzzy subset of a semihypergroup S. Then,A is an interval valued fuzzy left (right) hyperideal of S if and only if for each t ∈ D[0; 1], U(A; t) =∅ is a left (right) hyperideal of S, respectively. Proposition 2.9: LetA, B be interval valued fuzzy hyperideals of S, thenA ∧ B andA ∨ B are fuzzy hyperideals of S. Lemma 2.10: If A is an interval valued fuzzy left hyperideal and B is an interval valued fuzzy right hyperideal of S, thenA ◦ B is an interval valued fuzzy hyperideal of S andA ◦ B ≤ A ∧ B. 3. Interval Valued (α, β)-Fuzzy Bi-Hyperideals In this section, we give some useful characterizations of a semihypergroup in terms of interval valued (α, β)-fuzzy bi-hyperideals. Definition 3.1: Let A be an interval valued fuzzy set in a semihypergroup S. Then, A is called an interval valued fuzzy subsemihypergroup of S if for every z ∈ x ◦ y,inf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y)} for all x, y ∈ S. Definition 3.2: LetA be an interval valued fuzzy subsemihypergroup of a semihypergroup S. Then, A is called an interval valued fuzzy bi-hyperideal of S if for every w ∈ x ◦ y ◦ z, inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x), A(z)} for all x, y, z ∈ S. Theorem 3.3: For any interval valued fuzzy subset A of S, (B ) ⇔ (B ) and (B ) ⇔ (B ), 1 3 2 4 where (B ), (B ), (B ) and (B ) are given as follows: 1 2 3 4 (B ) inf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y)} for every x, y ∈ S. (B ) inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x), A(z)} for every x, y, z ∈ S. (B )(∀x, y ∈ S)(t, r ∈ D(0, 1])(x , y ∈ A ⇒ (z) ∈ A for every z ∈ x ◦ y). 3 r t min{t, r} (B )(∀x, y, z ∈ S)(t, r ∈ D(0, 1])(x , z ∈ A ⇒ (w) ∈ A for every w ∈ x ◦ y ◦ z). 4 r t min{t, r} Proof: (B ) ⇒ (B ).Let x, y ∈ S and t, r ∈ D(0, 1] be such that x , y ∈ A. Then, A(x) ≥ t 1 3 r and A(y) ≥ r.By (B ),wehaveinf A(z) ≥ min{A(x), A(y)}≥ min{t, r}. It follows that 1 z∈x◦y (z) ∈ A for every z ∈ x ◦ y. min{t, r} (B ) ⇒ (B ).Let x, y ∈ S.Since x ∈ A and y ∈ A.By (B ),wehave (z) 3 1 3 A(x) A(y) min{A(x),A(y)} ∈ A for every z ∈ x ◦ y. It follows that inf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y)}. (B ) ⇒ (B ).Let x, y, z ∈ S and t, r ∈ D(0, 1] be such that x , y ∈ A. Then, A(x) ≥ t and 2 4 t r A(z) ≥ r.By (B ),wehave ∈{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x), A(z)}≥ min{t, r. It follows that (w) ∈ A for every w ∈ x ◦ y ◦ z. min{t, r} 132 S. ABDULLAH ET AL. (B ) ⇒ (B ).Let x, y, z ∈ S.Since x ∈ A and z ∈ A.By (B ),wehave 4 2 4 A(x) A(z) (w) ∈ A for every w ∈ x ◦ y ◦ z. It follows that inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x),A(z)} min{A(x), A(z)}. Let A be an interval valued fuzzy subset of a semihypergroup S such that A(x) ≤ 0.5 for all x ∈ S.Let x ∈ S and t ∈ D(0, 1] be such that x ∈∧qA. Then, A(x) ≥ t and A(x) + t > 1. It follows that 1 < A(x) + t ≤ A(x) + A(x) = 2A(x), and so, A(x)> 0.5. This means {x |x ∈∧qA}=∅. t t Definition 3.4: Let S be a semihypergroups and A be an interval valued fuzzy set in S. Then A is called an interval valued (α, β)-fuzzy bi-hyperideal of S, where α =∈ ∧q, if for all x, y, z ∈ S and for all t, r ∈ D(0, 1] it satisfies: (B ) x , y αA ⇒ (z) βA for every z ∈ x ◦ y. 5 r t min{t, r} (B ) x , z αA ⇒ (w) βA for every w ∈ x ◦ y ◦ z. 6 r t min{t, r} Theorem 3.5: LetA be a non-zero interval valued (α, β)-fuzzy bi-hyperideal of S. Then, the set A :={x ∈ S : A(x)> 0} is a bi-hyperideal of S. Proof: Let x, y ∈ A . Then, A(x)> 0and A(y)> 0. Let us assume that A(z) = 0for every z ∈ x ◦ y.If α ∈{∈, ∈∨q}, then x αA and y αA. But for every z ∈ x ◦ y, A(x) A(y) (z) βA for every β ∈{∈, q, ∈∨q, ∈∧q}, which is a contradiction. Note that x qA min{A(x),A(y) and y qA. But for every z ∈ x ◦ y, (z) βA for every β ∈{∈, q, ∈∨q, ∈∧q}, which is a 1 min{1,1} contradiction. Hence, for every z ∈ x ◦ y, A(z)> 0, that is, z ∈ A for every z ∈ x ◦ y. Now, let x, z ∈ A and y ∈ S. Then, A(x)> 0and A(z)> 0. Let us suppose that A(w) = 0for every w ∈ x ◦ y ◦ z.If α ∈{∈, ∈∨q}, then x αA and z αA. But for every w ∈ x ◦ y ◦ A(x) A(z) z, (w) βA for every β ∈{∈, q, ∈∨q, ∈∧q}, which is a contradiction. Note that min{A(x),A(z)} x qA and z qA. But for every w ∈ x ◦ y ◦ z, (w) βA for every β ∈{∈, q, ∈∨q, ∈∧q}, 1 1 min{1,1} which is a contradiction. Hence, for every w ∈ x ◦ y ◦ z, A(w)> 0, that is, w ∈ A for every w ∈ x ◦ y ◦ z.Thus, A is a bi-hyperideal of S. Theorem 3.6: Let B be a bi-hyperideal of a semihypergroup S andA be an interval valued fuzzy subset of S such that (1) ∀ x ∈ S \ B, A(x) = 0, (2) ∀ x ∈ B, A(x) ≥ 0.5. Then, A is an interval valued (α, ∈∨q)-fuzzy bi-hyperideal of S. Proof: Case I: α = q.Let x, y ∈ S and r, t ∈ D(0, 1] be such that x qA and y qA. Then, x, y ∈ B and we have z ∈ B for every z ∈ x ◦ y.Ifmin{ r, t}≤ 0.5, then for every z ∈ x ◦ y,A(z) ≥ FUZZY INFORMATION AND ENGINEERING 133 0.5 ≥ min{ r, t} and hence (z) ∈ A.Ifmin{ r, t} > 0.5, then min{ r,t} A(z) + min{ r, t} > 0.5 + 0.5 = 1 for every z ∈ x ◦ y and so (z) qA. Therefore, (z) ∈∨qA for every z ∈ x ◦ y.Now,let x, y, z ∈ S and min{ r,t} min{ r,t} r, t ∈ D(0, 1] be such that x qA and z qA. Then, x, z ∈ B and we have w ∈ B for every w ∈ x ◦ y ◦ z.Ifmin{r, t}≤ 0.5, then for every w ∈ x ◦ y ◦ z, A(w) ≥ 0.5 ≥ min{r, t} and hence (w) ∈ A.Ifmin{ r, t} > 0.5, then min{ r,t} A(w) + min{ r, t} > 0.5 + 0.5 = 1 for every w ∈ x ◦ y ◦ z and so (w) qA. Therefore (w) ∈∨qA for every w ∈ x ◦ y ◦ z.Thus, A is an min{r,t} min{r,t} interval valued (q, ∈∨q)-fuzzy bi-hyperideal of S. Case II: α =∈.Let x, y ∈ S and r, t ∈ D(0, 1] be such that x ∈ A and y ∈ A. Then, x, y ∈ B r t andwehave z ∈ B for every z ∈ x ◦ y.Ifmin{ r, t}≤ 0.5, then for every z ∈ x ◦ y,suchthat A(z) ≥ 0.5 ≥ min{ r, t} and hence (z) ∈ A.Ifmin{ r, t} > 0.5, then min{ r,t} A(z) + min{ r, t} > 0.5 + 0.5 = 1 for every z ∈ x ◦ y and so (z) qA. Therefore, (z) ∈∨qA for every z ∈ x ◦ y.Now,let x, y, z ∈ S and min{ r,t} min{ r,t} r, t ∈ D(0, 1] be such that x ∈ A and z ∈ A. Then, x, z ∈ B andwehave w ∈ B for every w ∈ x ◦ y ◦ z.Ifmin{r, t}≤ 0.5, then for every w ∈ x ◦ y ◦ z, A(w) ≥ 0.5 ≥ min{r, t} and hence (w) ∈ A.Ifmin{r, t} > 0.5, then min{ r,t} A(w) + min{ r, t} > 0.5 + 0.5 = 1 for every w ∈ x ◦ y ◦ z and so (w) qA. Therefore, min{r,t} (w) ∈∨qA for every w ∈ x ◦ y ◦ z. min{ r,t} Thus, A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Case III: α =∈ ∨q. This follows from Cases I and II. Theorem 3.7: Every interval valued (∈, ∈)-fuzzy bi-hyperideal of a semihypergroup S is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal. Proof: The proof is straightforward. Theorem 3.8: Every interval valued (∈∨q, ∈∨q)-fuzzy bi-hyperideal of a semihypergroup S is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal. Proof: Let A be an interval valued (∈∨q, ∈∨q)-fuzzy bi-hyperideal of S.Let x, y ∈ S and t, r ∈ D(0, 1] be such that x qA and y qA. Then, x , y ∈∨qA, which implies (z) ∈∨qA r r t t min{ r,t} for every z ∈ x ◦ y.Now,let x, y, z ∈ S and t, r ∈ D(0, 1] be such that x ∈ A and z ∈ A. Then, x , z ∈∨qA, which implies that (w) ∈∨qA for every w ∈ x ◦ y ◦ z. t min{ r,t} 134 S. ABDULLAH ET AL. 4. (∈,∈∨q)-fuzzy Bi-hyperideals In this section, we define the notion of interval valued (∈, ∈∨q)-fuzzy bi-hyperideals of semihypergroups and investigate some of their properties in terms of interval valued (∈ , ∈∨q)-fuzzy bi-hyperideals. Definition 4.1: Let S be a semihypergroup and A be an interval valued fuzzy subset of S. Then, A is called an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S if (1) ∀ x, y ∈ S, ∀ t, r ∈ D(0, 1], x , y ∈ A ⇒ (z) ∈∨qA, for every z ∈ x ◦ y. t r min{t,r} (2) ∀ x, y, z ∈ S, ∀ t, r ∈ D(0, 1], x , z ∈ A ⇒ (w) ∈∨qA, for every w ∈ x ◦ y ◦ z. t r min{t, r} Definition 4.2: Let S be a semihypergroup and A be an interval valued fuzzy subset of S. Then, A is called an interval valued (∈, ∈∨q)-fuzzy left (resp. right) hyperideal of S if ∀x, y ∈ S, ∀t ∈ D(0, 1], y ∈ A ⇒ (z) ∈∨qA, for every z ∈ x ◦ y (resp. (z) ∈∨qA, for every t t t z ∈ y ◦ x). Theorem 4.3: Let S be a semihypergroup and A be an interval valued fuzzy subset of S. Then, A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S if and only if the following conditions hold: (1) inf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y), 0.5} for every x, y ∈ S. (2) inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x), A(z), 0.5} for every x, y, z ∈ S. Proof: Let A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S and let x, y ∈ S such that inf{A(z) : z ∈ x ◦ y} < min{A(x), A(y), 0.5}. Let us choose t ∈ D(0, 1] such that inf{A(z) : z ∈ x ◦ y} < t < min{A(x), A(y), 0.5}. Then we consider two cases: If min{A(x), A(y)} < 0.5, then inf{A(z) : z ∈ x ◦ y} < t < min{A(x), A(y)}. This implies that x , y ∈ A ⇒ (z) ∈∨qA for every z ∈ x ◦ y, which is a contradiction. t t t If min{A(x), A(y)}≥ 0.5, then x , y ∈ A, but (z) ∈∨qA, for every z ∈ x ◦ y,which is 0.5 0.5 0.5 a contradiction. Therefore, inf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y), 0.5}.Now,let x, y, z ∈ S. We consider the following cases: If min{A(x), A(z)} < 0.5, then assume that inf{A(w) : w ∈ x ◦ y ◦ z} < min{A(x), A(z), 0.5} for every x, y, z ∈ S. This implies that inf{A(w) : w ∈ x ◦ y ◦ z} < min{A(x), A(z)}.Letus choose t ∈ D(0, 1] such that inf{A(w) : w ∈ x ◦ y ◦ z} < t < min{A(x), A(z)}. Then x , z ∈ t t A, but (w) ∈∨qA, for every y ∈ S, which is a contradiction. If min{A(x), A(z)}≥ 0.5, then assume that inf{A(w) : w ∈ x ◦ y ◦ z} < 0.5 for every x, y, z ∈ S. Then x , y , z ∈ A, but (w) ∈∨qA, for every w ∈ x ◦ y ◦ z ∈ S, which is a 0.5 0.5 0.5 0.5 contradiction. Hence the condition (1) and (2) hold. Conversely, let us suppose that A satisfies the conditions (1) and (2). Let x, y ∈ S and t, r ∈ D(0, 1] such that x , y ∈ A. Then A(x) ≥ t, A(y) ≥ r.Wehaveinf{A(z) : z ∈ x ◦ y}≥ min{A(x), A(y), 0.5}≥ min{t, r, 0.5}.Ifmin{t, r} > 0.5, then inf{A(z) : z ∈ x ◦ y}≥ 0.5, FUZZY INFORMATION AND ENGINEERING 135 which implies that A(z) + min{t, r} > 1 for all z ∈ x ◦ y.Ifmin{t, r}≤ 0.5, then A(z) ≥ min{t, r}. Therefore, (z) ∈∨qA for all z ∈ x ◦ y. min{t, r} Let x, y, z ∈ S and t, r ∈ D(0, 1] such that x , y ∈ A. Then A(x) ≥ t, A(z) ≥ r.We have inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{A(x), A(z), 0.5}≥ min{t, r, 0.5} for every x, y, z ∈ S.If min{t, r} > 0.5, then inf{A(w) : w ∈ x ◦ y ◦ z}≥ 0.5, which implies that inf{A(w) : w ∈ x ◦ y ◦ z}+ min{t, r} > 1. If min{t, r}≤ 0.5, then inf{A(w) : w ∈ x ◦ y ◦ z}≥ min{t, r}. There- fore, (ω) ∈∨qA, for all w ∈ x ◦ y ◦ z. min{t, r} Example 4.4: Let (S, ·, ≤) be an ordered semigroup. Define a hyperoperation ° on S by: a ◦ b ={x ∈ S : x ≤ ab}= (ab] for all a, b ∈ S. Then for all a, b, c ∈ S,weclaimthat a ◦ (b ◦ c) = (a(bc)]. Let t ∈ a ◦ (b ◦ c), then t ∈ a ◦ x for some x ∈ (b ◦ c). This implies that t ≤ ax and x ≤ bc. Hence t ≤ a(bc),so a ◦ (b ◦ c) ⊆ (a(bc)]. Let s ∈ (a(bc)], then s ≤ a(bc).So s ∈ a◦⊆ (bc) ⊆ a ◦ x = a ◦ (b ◦ c). x∈b◦c Thus (a(bc)] ⊆ a ◦ (b ◦ c). Consequently, (a(bc)] = a ◦ (b ◦ c). Similarly, we can show that ((ab)c] = (a ◦ b) ◦ c. Hence (a ◦ b) ◦ c = a ◦ (b ◦ c).Thus (S, ◦) is a semihypergroup. Consider the ordered semihypergroup S ={a, b, c, d, e} with the following multiplication table and partial order relation: · ab c d e a ad ad d b aba d d c ad c d e d ad ad d e ad c d e ≤= {(a, a), (a, c), (a, d), (a, e), (b, b), (b, d), (b, e), (c, c), (c, e), (d, d), (d, e), (e, e)}. Then the hyperoperation ° is defined in the following table: ◦ ab c d e a a {a, b, d} a {a, b, d}{a, b, d} b ab a {a, b, d}{a, b, d} c a {a, b, d}{a, c}{a, b, d}{a, b, c, d, e} d a {a, b, d} a {a, b, d}{a, b, d} e a {a, b, d}{a, c}{a, b, d}{a, b, c, d, e} Then (S, ◦) is a semihypergroup and {a}, {a, b, e}, {a, b, d, e} are bi-hyperideals of S. We define an interval valued fuzzy subset A : S → D[0, 1] by: A(a) = [0.9, 0.8], A(b) = [0.75, 0.7], A(e) = [0.68, 0.65], A(d) = [0.45, 0.4], A(c) = [0.25, 0.2]. By routine calculation A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. But A is not one of the following interval valued fuzzy bi-hyperideals of S. (1) A is not a (q, ∈)-fuzzy bi-hyperideal of S, since a qA and b qA, but (a ◦ [0.78,0.73] [0.66,0.6] b) ={a, b, d} ∈ A. min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] 136 S. ABDULLAH ET AL. (2) A is not an (∈, ∈)-fuzzy bi-hyperideal of S, since a ∈ A and b ∈ A, but [0.78,0.73] [0.66,0.6] (a ◦ b) ={a, b, d} ∈ A. min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] (3) A is not an (∈, q)-fuzzy bi-hyperideal of S, since a ∈ A and b ∈ A, but [0.35,0.3] [0.25,0.2] (a ◦ b) ={a, b, d} qA. min{[0.35,0.3],[0.25,0.2]} [0.25,0.2] (4) A is not an (q, ∈∨q)-fuzzy bi-hyperideal of S, e qA and e qA, but (e ◦ [0.48,0.45] [0.48,0.45] e) ={a, b, c, d, e} ∈∨qA. min{[0.48,0.45],[0.48,0.45]} [0.48,0.45] (5) A is not a (q, ∈∧q)-fuzzy bi-hyperideal of S, since a qA and b qA, but [0.78,0.73] [0.66,0.6] (a ◦ b) ={a, b, d} ∈∧qA. min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] (6) A is not an (∈∨q, ∈∧q)-fuzzy bi-hyperideal of S, since a ∈∨qA and [0.78,0.73] b ∈∨qA, but (a ◦ b) ={a, b, d} ∈∧qA. [0.65,0.6] min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] (7) A is not an (∈∨q, ∈)-fuzzy bi-hyperideal of S, since a ∈∨qA and b ∈ [0.78,0.73] [0.66,0.6] ∨qA, but (a ◦ b) ={a, b, d} ∈ A. min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] (8) A is not an (∈, ∈∧q)-fuzzy bi-hyperideal of S, since a ∈ A and b ∈ A, [0.78,0.73] [0.66,0.6] but (a ◦ b) ={a, b, d} ∈ A and so {a, b, d} ∈∧qA. min{[0.78,0.73],[0.66,0.6]} [0.66,0.6] [0.66,0.6] (9) A is not a (q, q)-fuzzy bi-hyperideal of S, since a qA and b qA, but (a ◦ [0.38,0.35] [0.48,0.45] b) ={a, b, d} qA. min{[0.38,0.35],[0.48,0.45]} [0.38,0.35] (10) A is not an (∈∨q, q)-fuzzy bi-hyperideal of S, since a qA and b qA, but [0.38,0.35] [0.48,0.45] (a ◦ b) ={a, b, d} qA. min{[0.38,0.35],[0.48,0.45]} [0.38,0.35] (11) A is not an (∈∨q, ∈∨q)-fuzzy bi-hyperideal of S, since e ∈∨qA and [0.48,0.45] e ∈∨qA, but (e ◦ e) ={a, b, c, d, e} ∈∨qA. 0.48,0.45] min{[0.48,0.45],[0.48,0.45]} [0.48,0.45] From the above example (4), we see that an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal is not an interval valued (q, ∈∨q)-fuzzy bi-hyperideal. Example 4.5: Let S be the ordered semigroup of Example 4.4 and define an interval valued fuzzy subset A : S → D[0, 1] by: A(a) = [0.85, 0.8], A(b) = [0.75, 0.7], A(c) = [0.35, 0.3], A(d) = [0.55, 0.5], A(e) = [0.65, 0.6]. Thus, A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. But A is not an interval valued (α, β)-fuzzy bi-hyperideal of S. The following proposition holds. Proposition 4.6: Let A be a non-empty subset of a semihypergroup S. Then A is a bi-hyperideal of S if and only if the interval valued fuzzy set A of A is an interval valued (∈, ∈∨q)-fuzzy bi- hyperideal of S. In the following theorem we give a condition of being an interval valued (∈, ∈)-fuzzy bi-hyperideal of S. Theorem 4.7: LetA be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S such thatA(x)< 0.5 for all x ∈ S. Then A is an interval valued (∈, ∈)-fuzzy bi-hyperideal of S. FUZZY INFORMATION AND ENGINEERING 137 Proof: Let x, y ∈ S and r, t ∈ D(0, 1] be such that x ∈ A and y ∈ A. Then A(x) ≥ r and A(y) ≥ t,soforall z ∈ x ◦ y,wehave inf A(z) ≥ min{A(x), A(y), 0.5}≥ min{ r, t, 0.5}= min{ r, t}. z∈x◦y Hence, (z) ∈ A, for all z ∈ x ◦ y. min{ r,t} Now, let x, y, z ∈ S and r, t ∈ D(0, 1] be such that x ∈ A and z ∈ A. Then A(x) ≥ r and r t A(z) ≥ t and so for every w ∈ x ◦ y ◦ z,wehave inf A(w) ≥ min{A(x), A(z), 0.5}≥ min{ r, t, 0.5}= min{ r, t}. w∈x◦y◦z Consequently, (w) ∈ A for every w ∈ x ◦ y ◦ z.Thus, A is an interval valued (∈, ∈)- min{r,t} fuzzy bi-hyperideal of S. Theorem 4.8: Let S be a semihypergroup and A be an interval valued fuzzy subset of S. Then A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S if and only if U(A; t)(=∅) is a bi-hyperideal of S for all t ∈ D(0, 0.5]. Proof: Let A be an (∈, ∈∨q)-fuzzy bi-hyperideal of S and t ∈ D(0., 0.5]. Let x, y ∈ U(A; t). Then A(x) ≥ t and A(y) ≥ t. By hypothesis, we have inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}≥{t ∧ 0.5}= t, z∈x◦y and so z ∈ U(A; t) for every z ∈ x ◦ y.Thus x ◦ y ⊆ U(A; t). Hence U(A; t) is subsemihyper- group of S.Now,let x, y, z ∈ S such that for all t ∈ D(0, 0.5], x, z ∈ U(A; t). Then A(x) ≥ t and A(z) ≥ t. By hypothesis inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}≥{t ∧ 0.5}= t, w∈x◦y◦z and so, w ∈ U(A; t) for every w ∈ x ◦ y ◦ z.Thus x ◦ y ◦ z ⊆ U(A; t). Hence, U(A; t) is a bi- hyperideal of S. Conversely, let U(A; t) ={x ∈ S|A(x) ≥ t}(=∅) be a bi-hyperideal of S for all t ∈ D(0, 0.5]. Let x, y ∈ S. Then we have A(x) ≥{A(x) ∧ 0.5}= t and A(y) ≥{A(y) ∧ 0.5}= t ,so x, y ∈ U(A; t ).Since U(A; t ) is a bi-hyperideal of S,wehave x ◦ y ⊆ U(A; t ).Thus, 0 0 0 0 for every z ∈ x ◦ y,inf {A(z)}≥ t ={A(x) ∧ A(y) ∧ 0.5}.Now,let x, y, z ∈ S. Then z∈x◦y 0 we have A(x) ≥{A(x) ∧ 0.5}= t and A(z) ≥{A(z) ∧ 0.5}= t ,so x, z ∈ U(A; t ).Since 0 0 0 U(A; t ) is a bi-hyperideal of S,wehave x ◦ y ◦ z ⊆ U(A; t ). Thus, for every w ∈ x ◦ y ◦ 0 0 z,inf {A(w)}≥ t ={A(x) ∧ A(z) ∧ 0.5}. Hence, A is an interval valued (∈, ∈∨q)- w∈x◦y◦z 0 fuzzy bi-hyperideal of S. For an interval valued fuzzy subset A of a semihypergroup S and t ∈ D(0, 1], we denote Q(A; t) :={x ∈ S|x qA} and [A] :={x ∈ S|x ∈∨qA}. t t t Obviously, [A] = U(A; t) ∪ Q(A; t). We call [A] an (∈∨q)-level of A and Q(A; t) a q-level of A. t 138 S. ABDULLAH ET AL. We gave above a characterization of interval valued (∈, ∈∨q)-fuzzy bi-hyperideals by using these level subsets. In the following, we give another characterization of interval valued (∈, ∈∨q)-fuzzy bi-hyperideals by using the set [A] . Theorem 4.9: Let S be a semihypergroup and A be an interval valued fuzzy subset of S. Then, A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S if and only if [A] is a bi-hyperideal of S for all t ∈ D(0, 1]. Proof: Let us assume that A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S.Let x, y ∈ [A] for all t ∈ D(0, 1]. Then x ∈∨qA and y ∈∨qA, that is, A(x) ≥ t or A(x) + t > 1 t t t and A(y) ≥ t or A(y) + t > 1. Since A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S,wehave inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}. z∈x◦y Case 1. Let A(x) ≥ t and A(y) ≥ t.If t ≥ 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}= 0.5 z∈x◦y and hence (z) qA for every z ∈ x ◦ y.If t ≤ 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}≥ t z∈x◦y and so (z) ∈ A for every z ∈ x ◦ y. Hence (z) ∈∨qA for every z ∈ x ◦ y. t t Case 2. Let A(x) ≥ t and A(y) + t > 1. If t > 0.5, then inf ≥{A(x) ∧ A(y) ∧ 0.5} z∈x◦y ={A(y) ∧ 0.5} > {(1 − t) ∧ 0.5}, that is, for every z ∈ x ◦ y,wehave A(z) + t > 1and thus (z) qA for every z ∈ x ◦ y.If t ≤ 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}≥{t ∧ (1 − t) ∧ 0.5}= t z∈x◦y and so, (z) ∈ A for every z ∈ x ◦ y.Thus (z) ∈∨qA for every z ∈ x ◦ y. t t Case 3. Let A(x) + t > 1and A(y) ≥ t.If t > 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5} z∈x◦y ={A(x) ∧ 0.5} > {(1 − t) ∧ 0.5}= 1 − t, that is, for every z ∈ x ◦ y,wehave A(z) + t > 1and thus (z) qA for every z ∈ x ◦ y.If t ≤ 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}≥{(1 − t) ∧ (t ∧ 0.5}= t z∈x◦y and thus (z) ∈ A for every z ∈ x ◦ y. Hence, (z) ∈∨qA for every z ∈ x ◦ y. t t FUZZY INFORMATION AND ENGINEERING 139 Case 4. Let A(x) + t > 1and A(y) + t > 1. If t > 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5} z∈x◦y > {(1 − t) ∧ 0.5}= 1 − t, that is, for every z ∈ x ◦ y,wehave A(z) + t > 1and thus (z) qA for every z ∈ x ◦ y.If t ≤ 0.5, then inf {A(z)}≥{A(x) ∧ A(y) ∧ 0.5}≥{(1 − t) ∧ 0.5}= 0.5 ≥ t, z∈x◦y and so, (z) ∈ A for every z ∈ x ◦ y. Hence, in any case, we have (z) ∈∨qA for every z ∈ t t x ◦ y. Therefore, z ∈ [A] for every z ∈ x ◦ y. Now, let x, z ∈ [A] for all t ∈ D(0, 1] and y ∈ S. Then x ∈∨qA and z ∈∨qA, that is, t t t A(x) ≥ t or A(x) + t > 1and A(z) ≥ t or A(z) + t > 1. Since A is an interval valued (∈ , ∈∨q)-fuzzy bi-hyperideal of S,wehave inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}. w∈x◦y◦z Case 1. Let A(x) ≥ t and A(z) ≥ t.If t > 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}= 0.5 w∈x◦y◦z and hence (w) qA for every w ∈ x ◦ y ◦ z.If t ≤ 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}≥ t w∈x◦y◦z and so, (w) ∈ A for every w ∈ x ◦ y ◦ z. Hence (z) ∈∨qA for every w ∈ x ◦ y ◦ z. t t Case 2. Let A(x) ≥ t and A(z) + t > 1. If t > 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5} w∈x◦y◦z ={A(z) ∧ 0.5} > {(1 − t) ∧ 0.5}= 1 − t, that is, for every w ∈ x ◦ y ◦ z,wehave A(w) + t > 1and thus (w) qA for every w ∈ x ◦ y ◦ z.If t ≤ 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}≥{t ∧ (1 − t) ∧ 0.5}= t w∈x◦y◦z and so, (w) ∈ A for every w ∈ x ◦ y ◦ z.Thus (w) ∈∨qA for every w ∈ x ◦ y ◦ z. t t Case 3. Let A(x) + t > 1and A(z) ≥ t.If t > 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5} w∈x◦y◦z ={A(x) ∧ 0.5} > {(1 − t) ∧ 0.5}= 1 − t, 140 S. ABDULLAH ET AL. that is, for every w ∈ x ◦ y ◦ z,wehave A(w) + t > 1and thus (w) qA for every w ∈ x ◦ y ◦ z.If t ≤ 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}≥{(1 − t) ∧ t ∧ 0.5}= t w∈x◦y◦z and thus, (w) ∈ A for every w ∈ x ◦ y ◦ z. Hence (w) ∈∨qA for every w ∈ x ◦ y ◦ z. t t Case 4. Let A(x) + t > 1and A(z) + t > 1. If t > 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5} w∈x◦y◦z > {(1 − t) ∧ 0.5}= 1 − t, that is, for every w ∈ x ◦ y ◦ z,wehave A(w) + t > 1and thus (w) qA for every w ∈ x ◦ y ◦ z.If t ≤ 0.5, then inf {A(w)}≥{A(x) ∧ A(z) ∧ 0.5}≥{(1 − t) ∧ 0.5}= 0.5 ≥ t, w∈x◦y◦z and so (w) ∈ A for every w ∈ x ◦ y ◦ z. Hence, in any case, we have (w) ∈∨qA for every t t w ∈ x ◦ y ◦ z. Therefore, w ∈ [A] for every w ∈ x ◦ y ◦ z. Conversely, let A be an interval valued fuzzy subset of S and t ∈ D(0, 1] be such that [A] is a bi-hyperideal of S.Let x, y ∈ S be such that for every z ∈ x ◦ y,wehave A(z)< t < {A(x) ∧ A(y) ∧ 0.5} for some t ∈ D(0, 0.5]. Then x, y ∈ U(A; t) ⊆ [A] .This implies that for every z ∈ x ◦ y,wehave z ∈ [A] . Hence A(z) ≥ t or A(z) + t > 1for every z ∈ x ◦ y, which is a contradiction. Hence A(z) ≥{A(x) ∧ A(y) ∧ 0.5} for all x, y ∈ S. Now, let inf {A(w)} < {A(a) ∧ 0.5} for some a, x, y ∈ S. Choose t such that w∈x◦a◦y inf {A(w)} < t < {A(a) ∧ 0.5}. Then a ∈ U(A; t) ⊆ [A] . It follows that w ∈ [A] for w∈x◦a◦y t t every w ∈ x ◦ a ◦ y. This implies A(w) ≥ t or A(w) + t > 1 for every w ∈ x ◦ a ◦ y,which is a contradiction. Hence, inf {A(w)}≥{A(a) ∧ 0.5} for all a, x, y ∈ S. by Theorem 3.3, w∈x◦a◦y it follows that A is an interval valued (∈, ∈∨q)-fuzy bi-hyperideal of S. U(A; t) and [A] are bi-hyperideals of S, for all t ∈ D(0, 0.5], but Q(A; t) is not a bi- hyperideal of S, for all t ∈ D(0, 1], in general. The following example shows this. Example 4.10: Let S be the semihypergroup of Example 4.6. Define an interval valued fuzzy subset A by A(a) = [0.9, 0.8], A(b) = [0.75, 0.7], A(c) = [0.65, 0.6], A(d) = [0.4, 0.35], A(e) = [0.5, 0.45]. Then Q(A; t) ={a, b, c, e} for [0.4, 0.35] < t ≤ [0.5, 0.45]. Since a qA and b qA, [0.38,0.35] [0.32,0.3] but (c ◦ b) ={a, b, d} qA. Hence Q(A; t) is not a bi-hyperideal of S,for min{[0.38,0.35],[0.32,0.3]} 0.32 all t ∈ ([0.4, 0.35], [0.5, 0.45]). Definition 4.11 ([23]): Let S be a semihypergroup and A, B are interval valued fuzzy subsets of S. Then the 0.5-product of A and B is defined by: min{A(y), B(z), 0.5} if x ∈ y ◦ z (A ◦ B := x∈y◦z 0.5 0if x ∈ / y ◦ z. FUZZY INFORMATION AND ENGINEERING 141 We also define A B by (A B)(x) = min{A(x), B(x), 0.5} for all x ∈ S. 0.5 0.5 Proposition 4.12: If A and B are interval valued (∈, ∈∨q)-fuzzy bi-hyperideals of S, then A B is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. 0.5 Proof: It is straightforward. Lemma 4.13: Let S be a semihypergroup. Then, every interval valued (∈, ∈∨q)-fuzzy one sided hyperideal is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Proof: Let A be an interval valued (∈, ∈∨q)-fuzzy left hyperideal of S and x, y ∈ S. Then inf {A(z)}≥ min{A(y), 0.5}≥ min{A(x), A(y), 0.5}. z∈x◦y Hence, A is an interval valued (∈, ∈∨q)-fuzzy subsemihypergroup of S. Let x, y, z ∈ S. Then, for every α ∈ x ◦ y and for every w ∈ α ◦ z,wehave inf {A(w)}≥ min{A(z), 0.5}≥ min{A(x), A(z), 0.5}. w∈α◦z HenceA is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Similarly, we can prove that if A is an interval valued (∈, ∈∨q)-fuzzy right hyperideal of S, then A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Definition 4.14: An interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S is called idempotent if A ◦ A = A. 0.5 Proposition 4.15: Let S be a semihypergroup and A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Then A ◦ A ≤ A. 0.5 Proof: Let A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S.Let a, y, z ∈ S.If a ∈ / y ◦ z, then (A ◦ A)(a) = 0 ≤ A(a).If a ∈ y ◦ z, then 0.5 (A ◦ A)(a) = min{A(y), A(z), 0.5} 0.5 a∈y◦z ≤ A(a) = A(a). a∈y◦z For a semihypergroup S,the 0 (respectively 1) is defined as follows: (∀ x ∈ S)(0: S → D[0, 1]|x −→ 0(x) = 0 (∀ x ∈ S)(1: S → D[0, 1]|x −→ 1(x) = 1 Lemma 4.16: Let S be a semihypergroup and A, B be two interval valued fuzzy subsets of S. Then, A ◦ B ≤ 1 ◦ B (resp. A ◦ B ≤ A ◦ 1). 0.5 0.5 0.5 0.5 Proof: It is straightforward. 142 S. ABDULLAH ET AL. Proposition 4.17: Let S be a semihypergroup and A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. Then, A ◦ 1 ◦ A ≤ A. 0.5 0.5 Proof: Let a, y, z ∈ S.If a ∈ / y ◦ z, then (A ◦ 1 ◦ A)(a) = 0 ≤ A(a).If a ∈ y ◦ z, then 0.5 0.5 (A ◦ 1 ◦ A)(a) = min{A(y), (1 ◦ A)(z), 0.5} 0.5 0.5 0.5 a∈y◦z = min A(y), min{1(t), A(r), 0.5}, 0.5 for every z ∈ t ◦ r a∈y◦z z∈t◦r = min{A(y), 1, A(r), 0.5} a∈y◦z z∈t◦r = min{A(y), A(r), 0.5}= min{A(y), A(r), 0.5, 0.5}. a∈y◦z z∈t◦r a∈y◦z z∈t◦r Since a ∈ y ◦ z ⊆ y ◦ (t ◦ r) and A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S, we have A(a) ≥ min{A(y), A(r), 0.5}.Thus, min{A(y), A(r), 0.5, 0.5}≤ min{A(a), 0.5}= (A)(a), a∈y◦z a∈y◦z z∈t◦r z∈t◦r and consequently, (A ◦ 1 ◦ A)(a) ≤ (A)(a). 0.5 0.5 Theorem 4.18: A semihypergroup S is regular if and only if for every interval valued (∈, ∈∨q)- fuzzy bi-hyperideal A of S we have A ◦ 1 ◦ A = A where (A)(x) = min{A(x), 0.5}. 0.5 0.5 Proof: (⇒).Let S be a regular semihypergroup and let x ∈ S.Since S is regular, then there exists a ∈ S such that x ∈ x ◦ a ◦ x ⊆ x ◦ a ◦ (x ◦ a ◦ x) = x ◦ (a ◦ x ◦ a ◦ x). Then, for every t ∈ a ◦ x ◦ a and r ∈ t ◦ x,wehave x ∈ x ◦ r. Then we have (A ◦ 1 ◦ A)(x) = min{A(x), (1 ◦ A)(r), 0.5} 0.5 0.5 0.5 x∈x◦r ≥ min{A(x), (1 ◦ A)(r), 0.5} 0.5 = min A(x), min{1(t), A(a), 0.5}, 0.5 r∈t◦a ≥ min{A(x),min{1, A(a), 0.5}, 0.5} = min{A(x), 0.5}= (A)(a). Hence, (A)(a) ≤ (A ◦ 1 ◦ A)(a). On the other hand, by Proposition 4.20, we have 0.5 0.5 (A ◦ 1 ◦ A)(a) ≤ A(a). Therefore, (A ◦ 1 ◦ A)(a) = A(a). 0.5 0.5 0.5 0.5 (⇐).Let A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S such that the equality A ◦ 1 ◦ A = A is satisfied. To prove that S is regular, we will prove that B ◦ S ◦ B = B, 0.5 0.5 for all bi-hyperideal B of S.Let b ∈ B. Then by Proposition 4.8, we have A is an interval B FUZZY INFORMATION AND ENGINEERING 143 valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. By hypothesis, we have (A ◦ 1 ◦ A (b) = B B 0.5 0.5 A (b).Since b ∈ B, then A (b) = 1 and we have (A ◦ 1 ◦ A )(b) = 1. By Proposi- B B B B 0.5 0.5 tion 2.4, we have A ◦ 1 ◦ A = A and hence A (b) = 1 which implies that B B B◦S◦B B◦S◦B 0.5 0.5 b ∈ B ◦ S ◦ B.Thus B ⊆ B ◦ S ◦ B.Since B is a bi-hyperideal of S, then B ◦ S ◦ B ⊆ B. Therefore, B ◦ S ◦ B = B. Lemma 4.19: Let A and B be interval valued (∈, ∈∨q)-fuzzy bi-hyperideals of a semihyper- group S. Then A ◦ B is also an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S. 0.5 Proof: Let A and B be interval valued (∈, ∈∨q)-fuzzy bi-hyperideals of S and let a, y, z ∈ S. If a ∈ / y ◦ z, then ((A ◦ B) ◦ (A ◦ B)(a) = 0 ≤ (A ◦ B)(a).If a ∈ y ◦ z, then 0.5 0.5 0.5 0.5 ((A ◦ B) ◦ (A ◦ B)(a) 0.5 0.5 0.5 = {(A ◦ B)(y) ∧ (A ◦ B)(z) ∧ 0.5} 0.5 0.5 a∈y◦z = {A(p ) ∧ B(q ) ∧ 0.5}∧ {A(p ) ∧ B(q ) ∧ 0.5} 1 1 2 2 a∈y◦z y∈p ◦q z∈p ◦q 1 1 2 2 = {A(p ) ∧ B(q ) ∧ 0.5}∧{A(p ) ∧ B(q ) ∧ 0.5} 1 1 2 2 a∈y◦z y∈p ◦q z∈p ◦q 1 1 2 2 = {A(p ) ∧ A(p ) ∧ B(q ) ∧ B(q ) ∧ 0.5} 1 2 1 2 a∈y◦z y∈p ◦q z∈p ◦q 1 1 2 2 ≤ {A(p ) ∧ A(p ) ∧ B(q ) ∧ 0.5} . 1 2 2 a∈y◦z y∈p ◦q z∈p ◦q 1 1 2 2 Since a ∈ y ◦ z, y ∈ p ◦ q and z ∈ p ◦ q , then a ∈ (p ◦ q ) ◦ (p ◦ q ) = p ◦ (q ◦ p ◦ 1 1 2 2 1 1 2 2 1 1 2 q ) and so, (p , q ◦ p ◦ q ) ∈ A . Then we have 2 1 1 2 2 a [{A(p ) ∧ A(p ) ∧ B(q ) ∧ 0.5}]. 1 2 2 a∈y◦z y∈p ◦q z∈p ◦q 1 1 2 2 Since a ∈ y ◦ z, y ∈ p ◦ q and z ∈ p ◦ q , then a ∈ y ◦ z ⊆ (p ◦ q ) ◦ (p ◦ q ) = (p ◦ 1 1 2 2 1 1 2 2 1 q ◦ p ) ◦ q . So, for every p ∈ (p ◦ q ◦ p ), a ∈ p ◦ q . Then we have 1 2 2 1 1 2 2 a∈y◦z y∈p ◦q 1 1 [{A(p ) ∧ A(p ) ∧ B(q ) ∧ 0.5}] ≤ [{A(p ) ∧ A(p ) ∧ B(q ) ∧ 0.5}]. 1 2 2 1 2 2 z∈p ◦q a∈p◦q 2 2 2 Since A is interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S, then we have inf {A(p)}≥{A(p ) ∧ A(p ) ∧ 0.5}. 1 2 p∈p ◦q ◦p 1 1 2 Thus we have {A(p ) ∧ A(p ) ∧ B(q ) ∧ 0.5} 1 2 2 a∈p◦q ≤ {A(p) ∧ B(q ) ∧ 0.5} = (A ◦ B)(a). 0.5 a∈p◦q 2 144 S. ABDULLAH ET AL. Therefore, ((A ◦ B) ◦ (A ◦ B))(a) ≤ (A ◦ B)(a) and A ◦ B is an interval valued 0.5 0.5 0.5 0.5 0.5 (∈, ∈∨q)-fuzzy subsemihypergroup of S.Let x, y, z ∈ S. Then we have (A ◦ B)(x) ∧ (A ◦ B)(z) = {A(p) ∧ B(q) ∧ 0.5} ∧ {A(r) ∧ B(s) ∧ 0.5} 0.5 0.5 x∈p◦q z∈r◦s = {A(p) ∧ B(q) ∧ 0.5}∧{A(r) ∧ B(s) ∧ 0.5} x∈p◦q z∈r◦s = {A(p) ∧ A(r) ∧ B(q) ∧ B(s) ∧ 0.5} x∈p◦q z∈r◦s ≤ {A(p) ∧ A(r) ∧ B(s) ∧ 0.5} . x∈p◦q z∈r◦s Since x ∈ p ◦ q and z ∈ r ◦ s, then for every w ∈ x ◦ y ◦ z ⊆ (p ◦ q) ◦ y ◦ (r ◦ s) = (p ◦ (q ◦ y) ◦ r) ◦ s,wehave w ∈ (p ◦ (q ◦ y) ◦ r) ◦ s.Thus {A(p) ∧ A(r) ∧ B(s) ∧ 0.5} x∈p◦q z∈r◦s ≤ {A(p) ∧ A(r) ∧ B(s) ∧ 0.5} . w∈(p◦(q◦y)◦r)◦s Since A is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S, then for every α ∈ q ◦ y and for every β ∈ p ◦ α ◦ r,wehaveinf A(β) ≥{A(p) ∧ A(r) ∧ 0.5}. β∈p◦α◦r Hence, for every w ∈ β ◦ s ⊆ (p ◦ (q ◦ y) ◦ r) ◦ s,wehave {A(p) ∧ A(r) ∧ B(s) ∧ 0.5} w∈β◦s ≤ {A(β) ∧ B(s) ∧ 0.5} w∈β◦s = (A ◦ B)(w). 0.5 Thus (A ◦ B)(w) ≥ (A ◦ B)(x) ∧ (A ◦ B)(z). Therefore,A ◦ B is an interval valued 0.5 0.5 0.5 0.5 (∈, ∈∨q)-fuzzy bi-hyperideal of S. Theorem 4.20: Let S be a semihypergroup. The following statements are equivalent: (1) S is both regular and intra-regular. (2) A ◦ A = A for every interval valued (∈, ∈∨q)-fuzzy bi-hyperideal A of S. 0.5 (3) A B = (A ◦ B) (B ◦ A), for all interval valued (∈, ∈∨q)-fuzzy bi- 0.5 0.5 0.5 0.5 hyperideals A and B of S. Proof: (1) ⇒ (2).Let A be an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S and a ∈ S. Since S is regular and intra-regular, then there exist x, y, z ∈ S such that a ∈ a ◦ x ◦ a ⊆ a ◦ x ◦ a ◦ x ◦ a and a ∈ y ◦ a ◦ a ◦ z. Then a ∈ a ◦ x ◦ a ◦ x ◦ a ⊆ a ◦ x ◦ (y ◦ a ◦ a ◦ z) ◦ x ◦ a = (a ◦ (x ◦ y) ◦ a) ◦ (a ◦ (z ◦ x) ◦ a). Then for every r ∈ x ◦ y, s ∈ z ◦ x, p ∈ a ◦ r ◦ a and FUZZY INFORMATION AND ENGINEERING 145 q ∈ a ◦ s ◦ a.Thus a ∈ p ◦ q.Wehave (A ◦ A)(a) = {A(p) ∧ A(q) ∧ 0.5} 0.5 a∈p◦q ≥{A(p) ∧ A(q) ∧ 0.5} ≥ {A(a) ∧ A(a) ∧ 0.5}∧{A(a) ∧ A(a) ∧ 0.5}∧ 0.5 ={A(a) ∧ 0.5}= A(a). On the other hand, by Proposition 4.18, we have (A ◦ A)(a) ≤ A(a).Thus A ◦ A = A. 0.5 0.5 (2) ⇒ (3).Let A and B be interval valued (∈, ∈∨q)-fuzzy bi-hyperideals of S. Then A B is an interval valued (∈, ∈∨q)-fuzzy bi-hyperideal of S.By(2),wehave A B = 0.5 0.5 (A B) ◦ (A B) ⊆ A ◦ B. 0.5 0.5 0.5 0.5 Similarly, A B ⊆ B ◦ A.Thus A B ⊆ (A ◦ B) (B ◦ A). 0.5 0.5 0.5 0.5 0.5 0.5 On the other hand, by Lemma 4.22, A ◦ B and B ◦ A are interval valued (∈, ∈∨q)- 0.5 0.5 fuzzy bi-hyperideals of S. Hence A ◦ B B ◦ A is an interval valued (∈, ∈∨q)-fuzzy 0.5 0.5 0.5 bi-hyperideal of S. By (2), we have A ◦ B B ◦ A 0.5 0.5 0.5 ⎛ ⎞ ⎛ ⎞ ⎝ ⎠ ⎝ ⎠ = A ◦ B B ◦ A ◦ A ◦ B B ◦ A 0.5 0.5 0.5 0.5 0.5 0.5 0.5 ⊆ (A ◦ B) ◦ (B ◦ A) = A ◦ (B) ◦ (B) ◦ A 0.5 0.5 0.5 0.5 0.5 0.5 = A ◦ B ◦ A (by (1)) 0.5 0.5 ⊆ A ◦ 1 ◦ A 0.5 0.5 = A (by Theorem 4.21). By a similar way, we can prove that A ◦ B B ◦ A ⊆ B. Consequently, 0.5 0.5 0.5 A ◦ B B ◦ A ⊆ A B. 0.5 0.5 0.5 0.5 Therefore, A ◦ B B ◦ A = A B. 0.5 0.5 0.5 0.5 (3) ⇒ (1).Toprovethat S is regular, we prove that A ∩ B = A ◦ B ∩ B ◦ A for every bi-hyperideal A and B of S.Let x ∈ A ∩ B. By Proposition 4.8, A and A are interval val- A B ued (∈, ∈∨q)-fuzzy bi-hyperideals of S. By (3), A A )(x) = (A ◦ A A ◦ A B A B B 0.5 0.5 0.5 0.5 A )(x).Since x ∈ A and x ∈ B, then A (x) = 1and A (x) = 1. Then (A A )(x) = A A B A B 0.5 min{A (x), A (x), 0.5}= 0.5. Hence, (A ◦ A A ◦ A )(x) = 0.5. By Proposi- A B A B B A 0.5 0.5 0.5 tions 2.3 and 2.4, we haveA ◦ A A ◦ A = A and hence,A (x) = A B B A A◦B∩B◦A A◦B∩B◦A 0.5 0.5 0.5 0.5, which implies that x ∈ A ◦ B ∩ B ◦ A. On the other hand, if x ∈ A ◦ B ∩ B ◦ A, then 1 = A (x) A◦B∩B◦A ⎛ ⎞ ⎝ ⎠ = A A (x) A◦B B◦A 0.5 146 S. ABDULLAH ET AL. ⎛ ⎞ ⎝ ⎠ = A ◦ A A ◦ A (x) A B B A 0.5 0.5 0.5 ⎛ ⎞ ⎝ ⎠ = A A (x) (by (3)) A B 0.5 = A (x). A∩B Hence, x ∈ A ∩ B. Therefore, A ∩ B = A ◦ B ∩ B ◦ A. Consequently, S is both regular and intra-regular. This completes the proof. 5. Conclusions It is well-known that semihypergroups are basic structures in many applied branches of mathematics. Due to these posibilities of applications, semihypergroups are presently extensively investigated in fuzzy setting. On the other hand, interval valued fuzzy set the- ory emerges from the observation that in a number of cases, no objective procedure is available for selecting the crisp membership degrees of elements in a fuzzy set. It was suggested that problem by allowing to specify only an interval to which the actual mem- bership degree is assumed to belong. So, we combined these two concept to define a new generalization of fuzzy bi-hyperideals of semihypergroups. In this article, we defined inter- val valued (α, β)-fuzzy bi-hyperideals of semihypergroups. The concept of interval valued (α, β)-fuzzy bi-hyperideal is a generalization of the ordinary fuzzy bi-hyperideal, a gen- eralization of interval valued fuzzy bi-hyperideal and a generalization of interval valued (α, β)-fuzzy bi-hyperideal in semihypergroups. We studied fundamental results and also we provided some characterization theorems of interval valued (α, β)-fuzzy bi-hyperideals. We characterized some different classes of semihypergroups by the properties of interval valued (α, β)-fuzzy bi-hyperideals. In the future, we will focus on the following topics: (1) We will define n-dimensional fuzzy bi-hyperideals and characterize regular semihyper- groups by the properties of n-dimensional bi-hyperideals. (2) We will define n-dimensional (α, β)-fuzzy bi-hyperideals and characterize semi- groups by the properties of n-dimensional (α, β)-bi-hyperideals. (3) We will define interval valued intuitionistic fuzzy hyperideal with threshold ( α, β) in semihypergroups. (4) We will define n-dimensional intuitionistic fuzzy hyperideals and characterize semi- groups by the properties of n-dimensional intuitionistic fuzzy hyperideals. (5) We will study interval valued fuzzy hyperideals with threshold ( α, β) in other alge- braic hyperstructures i.e. LA-semihypergroups, hemirings, and hopefully we will obtain different results. Disclosure statement No potential conflict of interest was reported by the author(s). FUZZY INFORMATION AND ENGINEERING 147 Notes on contributors Saleem Abdullah received M. Phil and Ph.D. degrees in mathematics from Quaid-i-Azim University, Islamabad, Pakistan. Currently he is working as Assistant Professor at Department of Mathematics Abdul Wali Khan University, Mardan, Pakistan. His research area is fuzzy logic, fuzzy set theory and decision making. He has published more than 200 papers in national and international journals. Kostaq Hila is professor of Mathematics at The Department of Mathematical Engineering, Polytech- nic University of Tirana, Albania. He received his M.Sc. and PhD degree in Mathematics at University of Tirana, Albania. His main research interests include algebraic structures theory (in particular alge- braic theory of semigroups and ordered semigroups, LA-semigroups etc.), algebraic hyperstructures theory, fuzzy-rough-soft sets and applications. He has published several research papers in various international reputed peer-reviewed journals. He is a referee of several well-known international peer-reviewed journals. Shkelqim Kuka is professor of Mathematics at The Department of Mathematical Engineering, Poly- technic University of Tirana, Albania. He received his M.Sc. and PhD degree in Mathematics at Univer- sity of Tirana, Albania. His main research interests include algebraic structures theory, applications of fuzzy set theory, implementation and development of computer technology in data manage- ment and processing, implementation and applications of GIS systems etc. He has published several research papers in various international peer-reviewed journals. References [1] Zadeh LA. Fuzzy sets. Inform Control. 1965;8:338–353. [2] Rosenfeld A. Fuzzy groups. J Math Anal Appl. 1971;35:512–517. [3] Bhakat SK, Das P. (∈, ∈∨q)-fuzzy subgroups. Fuzzy Sets Syst. 1996;80:359–368. [4] Bhakat SK. (∈∨q)-level subset. Fuzzy Sets Syst. 1999;103:529–533. [5] Ahsan J, Li KY, Shabir M. Semigroups characterized by their fuzzy bi-ideals. J Fuzzy Math. 2002;10(2):441–449. [6] Ahsan J, Latif RM. Fuzzy quasi-ideals in semigroups. J Fuzzy Math. 2001;9(2):259–270. [7] Ahsan J, Saifullah K, Khan MF. Semigroups characterized by their fuzzy ideals. Fuzzy Syst Math. 1995;9:29–32. [8] Bhakat SK, Das P. On the denition of a fuzzy subgroup. 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(∈, ∈∨q)-fuzzy subnearrings and ideals. Soft Comput. 2006;10:206–211. [18] Corsini P. Join spaces, power sets, fuzzy sets. In: Proceedings of the 5th International Congress on Algebraic Hyperstructures and Applications; 1993; Isai, Romania. Hadronic Press; 1994. [19] Corsini P. New themes of research on hyperstructures associated with fuzzy sets. J Basic Sci. 2003;2(2):25–36. [20] Corsini P. A new connection between hypergroups and fuzzy sets. Southeast Bull Math. 2003;27:221–229. 148 S. ABDULLAH ET AL. [21] Corsini P, Shabir M, Mahmood T. Semisimple semihypergroups in terms of fuzzy hyperideals. Iran J Fuzzy Syst. 2011;8(1):95–111. [22] Leoreanu V. About hyperstructures associated with fuzzy sets of type 2. Ital J Pure Appl Math. 2005;17:127–136. [23] Tofan I, Volf AC. On some connections between hyperstructures and fuzzy sets. Ital J Pure Appl Math. 2000;7:63–68. [24] Kehagias A. Lattice-fuzzy meet and join hyperoperations. In: Proceedings of the 8th International Congress on AHA and Appl.; 2003; Samothraki, Greece. p. 171–182. [25] Cristea I. A property of the connection between fuzzy sets and hypergroupoids. Ital J Pure Appl Math. 2007;21:73–82. [26] Stefanescu M, Cristea I. On the fuzzy grade of hypergroups. Fuzzy Sets Syst. 2008;308:3537–3544. [27] Shabir M, Mahmood T. Semihypergroups characterized by (∈ , ∈ ∨q )-fuzzy hyperideals. J γ γ δ Intell Fuzzy Syst. 2015;28(6):2667–2678. [28] Jun YB, Khan A, Shabir M. Ordered semigroups characterized by their (∈, ∈∨q)-fuzzy bi-ideals. Bull Malays Math Sci Soc (2). 2009;32(3):391–408. [29] Jun YB, Öztürk MA, Yin Y. More general forms of generalized fuzzy bi-ideals in semigroups. Hacet J Math Stat. 2012;41(1):15–23. [30] Khan MSA, Abdullah S, Hila K. On the generalization of interval valued (∈, ∈∨q )-fuzzy general- ized bi-ideals in ordered semigroups. Appl Appl Math. In press. [31] Khan A, Jun YB, Shabir M. Ordered semigroups characterized by interval valued (∈, ∈∨q)-fuzzy bi-ideals. J Intell Fuzzy Syst. 2013;25(1):57–68. [32] Jun YB, Song SZ. Generalized fuzzy interior ideals in semigroups. Inform Sci. 2006;176:3079– [33] Jun YB. On (α, β)-fuzzy subalgebaras of BCK/BCI-algebras. Bull Korean Math Soc. 2005;42(4): 703–711. [34] Jun YB. Fuzzy subalgebras of type (α, β) in BCK/BCI-algebras. Kyungpook Math J. 2007;47: 403–410. [35] Kazanci O, Yamak S. Generalized fuzzy bi-ideals of semigroup. Soft Comput. 2008;12(11):1119– [36] Hasankhani A. Ideals in a semihypergroup and Greens relations. Ratio Math. 1999;13:29–36. [37] Zadeh LA. The concept of linguistic variable and its application to approximate reasoning-I. Inform Sci. 1975;8:199–249.
Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Apr 3, 2019
Keywords: Semihypergroups; interval valued -fuzzy bi-hyperideal; interval valued -fuzzy bi-hyperideals; 20N20; 08A72; 20N25