Generalized fuzzy sub-hyperquasigroups of ... - Springer Link

Soft Comput (2006) 10: 1109–1114 DOI 10.1007/s00500-006-0048-8

O R I G I NA L PA P E R

B. Davvaz · P. Corsini

Generalized fuzzy sub-hyperquasigroups of hyperquasigroups

Published online: 14 March 2006 © Springer-Verlag 2006

Abstract This paper concerns a relationship between fuzzy sets and algebraic hyperstructures. It is a continuation of ideas presented by Davvaz (Fuzzy Sets Syst 101: 191–195 1999) and Bhakat and Das (Fuzzy Sets Syst 80: 359-368 1996). In fact, the object of this paper is to study the notion of sub-hyperquasigroup in the (∈, ∈ ∨q)-fuzzy setting. Keywords Hyperquasigroup · Fuzzy set · Fuzzy logic · Implication operator 2000 Mathematics Subject Classification 20N20 · 20N25 · 03B52

1 Introduction In this section, we describe the motivation and a survey of related works. The theory of algebraic hyperstructures which is a generalization of the concept of ordinary algebraic structures first was introduced by Marty [34]. Since then many researchers have worked on algebraic hyperstructures and developed it. A short review of this theory appears in [10, 43]. A recent book [15] contains a wealth of applications. By this book, Corsini and Leoreanu (2003) presented some of the numerous applications of algebraic hyperstructures, especially those from the last fifteen years, to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence and probabilities. After the introduction of fuzzy sets by Zadeh [46], reconsideration of the concept of classical mathematics began. On B. Davvaz (B) Department of Mathematics, Yazd University, 89195-741 Yazd, Iran E-mail: [email protected] P. Corsini Dipartimento di Matematica e Informatica Via delle Scienze 206, 33100 Udine, Italy E-mail: [email protected]

the other hand, because of the importance of group theory in mathematics, as well as its many areas of application, the notion of fuzzy subgroup was defined by Rosenfeld [40] and its structure was investigated. This subject has been studied further in [1,3,6,16,31,33,36,37] and by many others [9,25, 29,30,35,39]. Das characterized fuzzy subgroups by their level subgroups in [16], since then many notions of fuzzy group theory can be equivalently characterized with the help of notion of level subgroups. In [3], Anthony and Sherwood redefined the fuzzy subgroup using the statistical functions. In [33], Liu introduced fuzzy sets in the realm of ring theory. A coherent study of the normal subgroups was initiated by Mukherjee and Bhattacharya [37]. Fuzzy quasinormality was introduced by Ajmal and Thomas [1]. The reader will find in [31,36,47], some basic definitions and theorems about the fuzzy sets and fuzzy algebra. A new type of fuzzy subgroup (viz, (∈, ∈ ∨q)-fuzzy subgroup) was introduced in an earlier paper of Bhakat and Das [6] by using the combined notions of “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets. In fact, (∈, ∈ ∨q)-fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. This concept has been studied further in [4,5,7,8]. Also, a generalization of Rosenfeld’s fuzzy subgroup, and Bhakat and Das’s fuzzy subgroup is given in [45]. Fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now used in the world both on the theoretical point of view and for their many applications. The relations between fuzzy sets and hyperstructures have been already considered by Corsini, Davvaz, Leoreanu, Zahedi, Ameri, Tofan, Kehagias and others, for instance see [2,11–13, 15,17–24,26–28,32,41,42,47]. In [17,18], Davvaz applied the concept of fuzzy sets to the theory of algebraic hyperstructures and defined fuzzy subhypergroup (resp. Hv -subgroup) of a hypergroup (resp. Hv -group) which is a generalization of the concept of Rosenfeld’s fuzzy subgroup of a group. In [24], Dudek, Davvaz and Jun, considered the intuitionistic fuzzification of the concept of sub-hyperquasigroups in a hyperquasigroup and investigated some properties of such hyperquasigroups, noting that a hypergroup (resp. Hv -group)

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is a hyperquasigroup with the associative (resp. weak associative) hyperoperation.

3 (∈, ∈ ∨q)-Fuzzy sub-hyperquasigroups A fuzzy subset A of H of the form t ( = 0) if y = x A(y) = 0 if y = x

2 Hyperquasigroups Hypergroupoid (H, ◦) is a non-empty set H with a hyperoperation ◦ defined on H , i.e. a mapping of H × H into the family of non-empty subsets of H . If (x, y) ∈ H × H, its image under ◦ is denoted by x ◦ y. If A, B ⊆ H then A ◦ B is given by A ◦ B = {x ◦ y | x ∈ A, y ∈ B}. x ◦ A is used for {x} ◦ A and A ◦ x for A ◦ {x}. Definition 2.1 A hypergroupoid (H, ◦) is called a hypergroup if for all x, y, z ∈ H the following two conditions hold: (i) x ◦ (y ◦ z) = (x ◦ y) ◦ z, (ii) x ◦ H = H ◦ x = H . The second condition is called the reproduciblity condition, means that for any x, y ∈ H there exist u, v ∈ H such that y ∈ x ◦ u and y ∈ v ◦ x. A hypergroupoid satisfying this condition is called a hyperquasigroup. Thus a hypergroup is a hyperquasigroup with the associative hyperoperation. A non-empty subset K of a hyperquasigroup (H, ◦) is called a sub-hyperquasigroup if (K , ◦) is a hyperquasigroup. Davvaz applied in [17,18] fuzzy sets to the theory of algebraic hyperstructures and studied their fundamental properties. Further investigations are contained in [19–24]. Definition 2.2 (cf. [18]) Let (H, ◦) be a hypergroup (resp. hyperquasigroup) and let A a fuzzy subset of H . Then A is said to be a fuzzy sub-hypergroup (resp. sub-hyperquasigroup) of H if the following axioms hold: 1. A(x) ∧ A(y) ≤ A(z) for all x, y ∈ H , z∈x◦y

2. for all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and A(a) ∧ A(x) ≤ A(y), 3. for all x, a ∈ H there exists z ∈ H such that x ∈ z ◦ a and A(a) ∧ A(x) ≤ A(z). For any fuzzy set A in H and any t ∈ (0, 1], we define the set At = {x ∈ H | A(x) ≥ t}, which is called a t-level cut of A. Theorem 2.3 (cf. [18]) Let H be a hyperquasigroup and A a fuzzy set of H . Then A is a fuzzy sub-hyperquasigroup of H if and only if for every t ∈ (0, 1], At ( = ∅) is a sub-hyperquasigroup of H . When A is a fuzzy sub-hyperquasigroup of H , At is called a level sub-hyperquasigroup of H . The concept of level subhyperquasigroups has been used extensively to characteristic various properties of fuzzy sub-hyperquasigroups.

is said to be fuzzy point with support x and value t and is denoted by x t . A fuzzy point xt is said to belong to (resp. be quasi-coincident with) a fuzzy set A, written as xt ∈ A (resp. xt q A) if A(x) ≥ t (resp. A(x) + t > 1). If xt ∈ A or xt q A, then we write xt ∈ ∨q A. The symbol ∈, ∨q means either ∈ or q hold. Based on [7], we can extend the concept of (∈, ∈ ∨q)fuzzy subgroups to the concept of (∈, ∈ ∨q)-fuzzy subhyperquasigroups in the following way: Definition 3.1 A fuzzy subset A of a hyperquasigroup H is said to be an (∈, ∈ ∨q)-fuzzy sub-hyperquasigroup of H if for all t, r ∈ (0, 1] and x, y ∈ H , (i) xt , yr ∈ A implies z t∧r ∈ ∨q A for all z ∈ x ◦ y; (ii) xt , ar ∈ A implies yt∧r ∈ ∨q A for some y ∈ H with x ∈ a ◦ y; (iii) xt , ar ∈ A implies z t∧r ∈ ∨q A for some z ∈ H with x ∈ z ◦ a. Proposition 3.2 Conditions (i), (ii) and (iii) in Deﬁnition 3.1, are equivalent to the following conditions respectively. 1. A(x) ∧ A(y) ∧ 0.5 ≤ A(z) for all x, y ∈ H ; z∈x◦y

2. for all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and A(a) ∧ A(x) ∧ 0.5 ≤ A(y); 3. for all x, a ∈ H there exists z ∈ H such that x ∈ z ◦ a and A(a) ∧ A(x) ∧ 0.5 ≤ A(z). Proof (i ⇒ 1): Suppose that x, y ∈ H . We consider the following cases: (a) A(x) ∧ A(y) < 0.5, (b) A(x) ∧ A(y) ≥ 0.5. Case a Assume that there exists z ∈ x ◦y such that A(z) < A(x) ∧ A(y) ∧ 0.5, which implies A(z) < A(x) ∧ A(y). Choose t such that A(z) < t < A(x)∧ A(y). Then xt , yt ∈ A, but z t ∈ ∨q A which contradicts (i). Case b Assume that A(z) < 0.5 for some z ∈ x ◦ y. Then x0.5 , y0.5 ∈ A, but z 0.5 ∈ ∨q A, a contradiction. Hence (1) holds. (ii ⇒ 2): Suppose that x, a ∈ H . We consider the following cases: (a) A(x) ∧ A(a) < 0.5, (b) A(x) ∧ A(a) ≥ 0.5. Case a Assume that for all y with x ∈ a ◦ y, we have A(y) < A(x) ∧ A(a). Choose t such that A(y) < t < A(x)∧ A(a) and t+ A(y) < 1. Then xt , at ∈ A, but yt ∈ ∨q A, which contradicts (ii).


Case b Assume that for all y with x ∈ a ◦ y, we have A(y) < A(x) ∧ A(a) ∧ 0.5. Then x0.5 , a0.5 ∈ A, but y0.5 ∈ ∨q A, which contradicts (ii). Hence (2) holds. (iii ⇒ 3): The proof is similar to (ii ⇒ 2). (1 ⇒ i): Let xt , yr ∈ A, then A(x) ≥ t and A(y) ≥ r . For every z ∈ x ◦ y we have A(z) ≥ A(x) ∧ A(y) ∧ 0.5 ≥ t ∧ r ∧ 0.5. If t ∧ r > 0.5, then A(z) ≥ 0.5 which implies A(z) + t ∧ r > 1. If t ∨ r ≤ 0.5, then A(z) ≥ t ∧ r . Therefore z t∧r ∈ ∨q A for all z ∈ x ◦ y. (2 ⇒ ii): Let xt , ar ∈ A, then A(x) ≥ t and A(a) ≥ r . Now, for some y with x ∈ a ◦ y we have A(y) ≥ A(a) ∧ A(x) ∧ 0.5 ≥ t ∧ r ∧ 0.5. If t ∧ r > 0.5, then A(y) ≥ 0.5 which implies A(y) + t ∧ r > 1. If t ∨ r ≤ 0.5, then A(y) ≥ t ∧ r . Therefore yt∧r ∈ ∨q A. Hence (ii) holds. (3 ⇒ iii): The proof is similar to (2 ⇒ ii).

By Definition 3.1 and Proposition 3.2, we immediately get: Corollary 3.3 A fuzzy subset A of a hyperquasigroup H is an (∈, ∈ ∨q)-fuzzy sub-hyperquasigroup of H if and only if the conditions (1), (2) and (3) in Proposition 3.2 hold. Now, we characterize (∈, ∈ ∨q)-fuzzy subhyperquasigroups by their level sub-hyperquasigroups. Theorem 3.4 Let A be a fuzzy sub-hyperquasigroup of H . Then for all 0 < t ≤ 0.5, At is an empty set or a sub-hyperquasigroup of H . Conversely, if A is a fuzzy subset of H such that At ( = ∅) is a sub-hyperquasigroup of H for all 0 < t ≤ 0.5, then A is a (∈, ∈ ∨q)-fuzzy sub-hyperquasigroup of H . Proof Let A be a fuzzy sub-hyperquasigroup of H and 0 < t ≤ 0.5. Let x, y ∈ At . Then A(x) ≥ t and A(y) ≥ t. Now A(z) ≥ A(x) ∧ A(y) ∧ 0.5 ≥ t ∧ 0.5 = t. z∈x◦y

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then x ∈ At0 and y ∈ At0 , so x ◦ y ⊆ At0 . Therefore for every z ∈ x ◦ y we have A(z) ≥ t0 which implies A(z) ≥ t0 , z∈x◦y

and in this way the condition (1) of Proposition 3.2 is verified. To verify the second condition, if for every a, x ∈ H , we put t1 = A(a) ∧ A(x) ∧ 0.5, then x ∈ At1 and a ∈ At1 . So there exists y ∈ At1 such that x ∈ a ◦ y. Since y ∈ At1 , we have A(y) ≥ t1 or A(y) ≥ A(a) ∧ A(x) ∧ 0.5. The third condition is verified similarly.

Naturally, a corresponding result should be considered when At is a sub-hyperquasigroup of H for all t ∈ (0.5, 1]. Theorem 3.5 Let A be a fuzzy subset of a hyperquasigroup H . Then At ( = ∅) is a sub-hyperquasigroup of H for all t ∈ (0.5, 1] if and only if 1. A(x) ∧ A(y) ≤ (A(z) ∨ 0.5) for all x, y ∈ H ; z∈x◦y

2. for all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and A(a) ∧ A(x) ≤ A(y) ∨ 0.5; 3. for all x, a ∈ H there exists z ∈ H such that x ∈ z ◦ a and A(a) ∧ A(x) ≤ A(z) ∨ 0.5. Proof ( ⇒): If there exist x, y, z ∈ H with z ∈ x ◦ y such that A(z) ∨ 0.5 < A(x) ∧ A(y) = t, then t ∈ (0.5, 1], A(z) < t, x ∈ At , and y ∈ At . Since x, y ∈ At and At is a sub-hyperquasigroup, so x ◦ y ⊆ At and A(z) ≥ t for all z ∈ x ◦ y, which is a contradiction with A(z) < t. Therefore A(x)∧ A(y) ≥ A(z) ∨ 0.5 for all x, y, z ∈ H with z ∈ x ◦ y, which implies A(x) ∧ A(y) ≥

(A(z) ∨ 0.5) for all x, y ∈ H.

Therefore for every z ∈ x ◦ y we have A(z) ≥ t or z ∈ At , so x ◦ y ⊆ At . Hence for every a ∈ At we have a ◦ At ⊆ At . Now, let x, a ∈ At , then there exists y ∈ H such that x ∈ a◦y and A(a) ∧ A(x) ∧ 0.5 ≤ A(y). From x, a ∈ At , we have A(x) ≥ t and A(a) ≥ t, and so

Hence (1) holds. Now, assume that there exist x0 , a0 ∈ H such that for all y ∈ H with x0 ∈ a0 ◦ y, the following inequality holds:

t = t ∧ t ∧ 0.5 ≤ A(a) ∧ A(x) ∧ 0.5 ≤ A(y).

A(y) ∨ 0.5 < A(a0 ) ∧ A(x0 ) = t.

Hence y ∈ At , and this prove that At ⊆ a ◦ At . Conversely, let A be a fuzzy subset of H such that At ( = ∅) is a sub-hyperquasigroup of H for all 0 < t ≤ 0.5. For every x, y ∈ H , we can write

Then t ∈ (0.5, 1], x0 ∈ At , a0 ∈ At and A(y) < t. Since x0 , a0 ∈ At and At is a sub-hyperquasigroup, so there exists y0 ∈ At such that x0 ∈ a0 ◦ y0 . From y0 ∈ At , we get A(y0 ) ≥ t, which is a contradiction with A(y0 ) < t. Therefore for all x, a ∈ H there exists y ∈ H such that x ∈ a◦y and

A(x) ≥ A(x) ∧ A(y) ∧ 0.5 = t0 , A(y) ≥ A(x) ∧ A(y) ∧ 0.5 = t0 ,

z∈x◦y

A(a) ∧ A(x) ≤ A(y) ∨ 0.5.

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Hence (2) holds. The proof of third condition is similar to the proof of second condition. (⇐ ): Assume that t ∈ (0.5, 1] and x, y ∈ At . Then (A(z) ∨ 0.5). 0.5 < t ≤ A(x) ∧ A(y) ≤

Proof Let A be a fuzzy sub-hyperquasigroup with thresholds of H and t ∈ (α, β]. Let x, y ∈ At . Then A(x) ≥ t and A(y) ≥ t. Now (A(z) ∨ α) ≥ A(x) ∧ A(y) ∧ β ≥ t ∧ β ≥ t > α.

z∈x◦y

It follows that for every z ∈ x ◦ y, 0.5 < t ≤ A(z) ∨ 0.5 and so t ≤ A(z), which implies z ∈ At . Hence x ◦ y ⊆ At . Now, let x, a ∈ At . Then using condition (2), there exists y ∈ H such that x ∈ a ◦ y and

So for every z ∈ x ◦ y we have A(z) ∨ α ≥ t > α which implies A(z) ≥ t and z ∈ At . Hence x ◦ y ⊆ At . Now, let x, a ∈ At , then there exists y ∈ H such that x ∈ a ◦ y and A(a) ∧ A(x) ∧ β ≤ A(y) ∨ α. From x, a ∈ At , we have A(x) ≥ t and A(a) ≥ t, and so

A(a) ∧ A(x) ≤ A(y) ∨ 0.5.

α < t ≤ t ∧ β ≤ A(a) ∧ A(x) ∧ β ≤ A(y) ∨ α,

We show that y ∈ At . We have

which implies A(y) ≥ t, and so y ∈ At . Therefore we have At = a ◦ At for all a ∈ At . Similarly we get At ◦ a = At for all a ∈ At . Therefore At is a sub-hyperquasigroup of H for all t ∈ (α, β]. Conversely, let A be a fuzzy subset of H such that At ( = ∅) is a sub-hyperquasigroup of H for all t ∈ (α, β]. If there exist x, y, z ∈ H with z ∈ x ◦ y such that

0.5 < t ≤ A(x) ≤ A(a) ∧ A(x) ≤ A(y) ∨ 0.5. It follows that 0.5 ≤ A(y) and so y ∈ At . Therefore At is a sub-hyperquasigroup of H for all t ∈ (0.5, 1].

Let A be a fuzzy subset of a hyperquasigroup H and J = {t | t ∈ (0, 1] and At is an empty-set or a sub-hyperquasigroup of H }. When J = (0, 1], then A is an ordinary fuzzy sub-hyperquasigroup of the hyperquasigroup H (Theorem 2.3). When J = (0, 0.5], A is an (∈, ∈ ∨q)-fuzzy sub-hyperquasigroup of the hyperquasigroup H (Theorem 3.4). In [45], Yuan, Zhang and Ren gave the definition of a fuzzy subgroup with thresholds which is a generalization of Rosenfeld’s fuzzy subgroup, and Bhakat and Das’s fuzzy subgroup. Based on [45], we can extend the concept of a fuzzy subgroup with thresholds to the concept of fuzzysub-hyperquasigroup with thresholds in the following way: Definition 3.6 Let α, β ∈ [0, 1] and α < β. Let A be a fuzzy subset of a hyperquasigroup H . Then A is called a fuzzy sub-hyperquasigroup with thresholds (α, β) of H if for all x, y ∈ H , 1. A(x) ∧ A(y) ∧ β ≤ z∈x◦y (A(z) ∨ α) for all x, y ∈ H ; 2. for all x, a ∈ H there exists y ∈ H such that x ∈ a ◦ y and A(a) ∧ A(x) ∧ β ≤ A(y) ∨ α; 3. for all x, a ∈ H there exists z ∈ H such that x ∈ z ◦a and A(a) ∧ A(x) ∧ β ≤ A(z) ∨ α.

z∈x◦y

A(z) ∨ α < A(x) ∧ A(y) ∧ β = t, then t ∈ (α, β], A(z) < t, x ∈ At and y ∈ At . Since At is a sub-hyperquasigroup of H and x, y ∈ At , so x ◦ y ⊆ At . Hence A(z) ≥ t for all z ∈ x ◦ y. This is a contradiction with A(z) < t. Therefore A(x)∧ A(y)∧β ≤ A(z)∨α for all x, y, z ∈ H with z ∈ x ◦ y, which implies A(x) ∧ A(y) ∧ β ≤

(A(z) ∨ α) for all x, y ∈ H.

z∈x◦y

Hence condition (1) of Definition 3.6 holds. Now, assume that there exist x0 , a0 ∈ H such that for all y ∈ H which satisfies x0 ∈ a0 ◦ y, the following inequality holds: A(y) ∨ α < A(a0 ) ∧ A(x0 ) ∧ β = t. Then t ∈ (α, β], x0 ∈ At , a0 ∈ At and A(y) < t. Since x0 , a0 ∈ At and At is a sub-hyperquasigroup, so there exists y0 ∈ At such that x0 ∈ a0 ◦ y0 . From y0 ∈ At , we get A(y0 ) ≥ t. This is a contradiction with A(y0 ) < t. Therefore A(a) ∧ A(x) ∧ β ≤ A(y) ∨ α. Hence the second condition of Definition 3.6 holds. The proof of third condition is similar to the proof of second condition.

If A is a fuzzy sub-hyperquasigroup with thresholds of H , then we can conclude that A is an ordinary fuzzy sub-hyperquasigroup when α = 0, β = 1; and A is an (∈, ∈ ∨q)-fuzzy sub-hyperquasigroup when α = 0, β = 0.5. Now, we characterize fuzzy sub-hyperquasigroups with thresholds by their level sub-hyperquasigroups.

4 Implication-based fuzzy sub-hyperquasigroup

Theorem 3.7 A fuzzy subset A of a hyperquasigeoup H is a fuzzy subhyperquasigroup with thresholds (α, β) of H if and only if At ( = ∅) is a sub-hyperquasigroup of H for all t ∈ (α, β].

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables (or terms of the linguistic variable truth). Some operators, like ∧, ∨, ¬, −→ in fuzzy logic are also defined by using truth tables,


the extension principle can be applied to derive definitions of the operators. In fuzzy logic, truth value of fuzzy proposition P is denoted by [P]. In the following, we display the fuzzy logical and corresponding set-theoretical notions. [x ∈ A] = A(x), [x ∈ A] = 1 − A(x), [P ∧ Q] = min{[P], [Q]}, [P ∨ Q] = max{[P], [Q]}, [P −→ Q] = min{1, 1 − [P] + [Q]}, [∀x P(x)] = inf[P(x)], | P if and only if [P] = 1 for all valuations. Of course, various implication operators have been defined. We only show a selection of them in the next table. α denotes the degree of truth (or degree of membership) of the premise, β the respective values for the consequence, and I the resulting degree of truth for the implication.

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Definition 4.2 A fuzzy subset A of a hyperquasigroup H is called a t-implication-based fuzzy sub-hyperquasigroup of H with respect to the implication −→ iff it satisfies: 1. for any x, y ∈ H , | t [ [x ∈ A] ∧ [y ∈ A] −→ [∀z ∈ x ◦ y, z ∈ A] ], 2. for any x, a ∈ H there exists y ∈ H with x ∈ a ◦ y, | t [ [x ∈ A] ∧ [a ∈ A] −→ [y ∈ A]], 3. for any x, a ∈ H there exists z ∈ H with x ∈ z ◦ a, | t [ [x ∈ A] ∧ [a ∈ A] −→ [z ∈ A]]. Now, let I be an implication operator. Then Corollary 4.3 A fuzzy subset A of a hyperquasigroup H is a t-implication-based fuzzy sub-hyperquasigroup of H with respect to the implication I iff (i) I A(x) ∧ A(y), A(z) ≥ t for all x, y ∈ H , z∈x◦y

Name Early Zadeh Lukasiewicz

Definition of implication operator Im (α, β) = max{1−α, min{α, β}} Ia (α, β) = min{1, 1 − α + β} 1 α≤β Standard Star (Godel) Ig (α, β) = β elsewhere 1 α≤β Contraposition of Godel Icg (α, β) = 1 − α elsewhere 1α≤β Gaines-Rescher Igr (α, β) = 0 elsewhere Kleene-Dienes Ib (α, β) = max{1 − α, β}

The “quality” of these implication operators could be evaluated either empirically or axiomatically. In the following definition we considered the definition of implication operator in the Lukasiewicz system of countinuous-valued logic. Definition 4.1 A fuzzy subset A of a hyperquasigroup H is called a fuzzifying sub-hyperquasigroup of H iff it satisfies: 1. for any x, y ∈ H , | [ [x ∈ A] ∧ [y ∈ A] −→ [∀z ∈ x ◦ y, z ∈ A] ], 2. for any x, a ∈ H there exists y ∈ H with x ∈ a ◦ y, | [ [x ∈ A] ∧ [a ∈ A] −→ [y ∈ A]], 3. for any x, a ∈ H there exists z ∈ H with x ∈ z ◦ a, | [ [x ∈ A] ∧ [a ∈ A] −→ [z ∈ A]]. Clearly, Definition 4.1 is equivalent to Definition 2.2. Therefore, a fuzzifying sub-hyperquasi group is an ordinary fuzzy sub-hyperquasigroup. In [44], the concept of t-tautology is introduced, i.e., | t P if and only if [P] ≥ t for all valuations. Based on [45], we can extend the concept of implicationbased fuzzy subgroup to the concept of implication-based fuzzy sub-hyperquasigroup in the following way:

(ii) for any x, a ∈ H there exists y ∈ H with x ∈ a ◦ y and I (A(x) ∧ A(a), A(y)) ≥ t, (iii) for any x, a ∈ H there exists z ∈ H with x ∈ z ◦ a and I (A(x) ∧ A(a), A(z)) ≥ t. Let A be a fuzzy subset of a hyperquasigroup H , then we have the following results: Theorem 4.4 1. Let I = Igr . Then A is an 0.5-implication-based fuzzy sub-hyperquasigroup of H if and only if A is a fuzzy subhyperquasigroup with thresholds α = 0 and β = 1 of H. 2. Let I = Ig . Then A is an 0.5-implication-based fuzzy sub-hyperquasigroup of H if and only if A is a fuzzy sub-hyperquasigroup with thresholds α = 0 and β = 0.5 of H. 3. Let I = Icg . Then A is an 0.5-implication-based fuzzy sub-hyperquasigroup of H if and only if A is a fuzzy sub-hyperquasigroup with thresholds α = 0.5 and β = 1 of H. Proof The proof is straightforward, by considering the definitions.

Acknowledgements The first author would like to thank the research council of the Yazd University for financial support.

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