A New View on Fuzzy Hypermodules

Acta Mathematica Sinica, English Series Aug., 2007, Vol. 23, No. 8, pp. 1345–1356 Published online: June 20, 2007 DOI: 10.1007/s10114-007-0951-7 Http://www.ActaMath.com

A New View on Fuzzy Hypermodules Jian Ming ZHAN Department of Mathematics, Hubei Institute for Nationalities, Enshi 445000, P. R. China E-mail: [email protected]

Bijan DAVVAZ Department of Mathematics, Yazd University, Yazd, Iran E-mail: [email protected]

K. P. SHUM Faculty of Science, The Chinese University of Hong Kong, Shatin, Hong Kong (SAR), P. R. China E-mail: [email protected] Abstract We describe the relationship between the fuzzy sets and the algebraic hyperstructures. In fact, this paper is a continuation of the ideas presented by Davvaz in (Fuzzy Sets Syst., 117: 477– 484, 2001) and Bhakat and Das in (Fuzzy Sets Syst., 80: 359–368, 1996). The concept of the quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set is introduced and this is a natural generalization of the quasi-coincidence of a fuzzy point in fuzzy sets. By using this new idea, the concept of interval-valued (α, β)-fuzzy sub-hypermodules of a hypermodule is defined. This newly defined interval-valued (α, β)-fuzzy sub-hypermodule is a generalization of the usual fuzzy sub-hypermodule. We shall study such fuzzy sub-hypermodules and consider the implication-based interval-valued fuzzy sub-hypermodules of a hypermodule. Keywords hypermodule, interval-valued (α, β)-fuzzy sub-hypermodule, interval-valued (∈, ∈ ∨q)fuzzy sub-hypermodule, fuzzy logic, implication operator MR(2000) Subject Classification 20N20, 20N25, 03B52

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Introduction

Hyperstructure theory was first initiated by Marty [1] in 1934 when he defined hypergroups and started to analyse their properties. Because there are extensive applications in many branches of mathematics and applied sciences, the theory of algebraic hyperstructures (or hypersystems) has nowadays become a well-established branch in algebraic theory. Later on, people have developed the semi-hypergroups, which are the simplest algebraic hyperstructures having closure and associativity properties. A short review of the theory of hyperstructures can be found in [2–4]. The recent monograph of Corsini and Leoreanu [5] also contains a rich source of applications. In fact, Corsini and Leoreanu [5] presented numerous applications of algebraic hyperstructures, especially those from the last decade, including the applications of hyperstructures in geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence and probabilities. Received May 25, 2006, Accepted January 12, 2007 The research of the first author is partially supported by the National Natural Science Foundation of China (60474022) and the Key Science Foundation of Education Commission of Hubei Province, China (D200729003; D200529001) and the research of the third author is partially supported by an RGC grant (CUHK) #2060297 (05/07)

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After introducing the celebrated concept of fuzzy sets, people have tried to implement the fuzzy concept of Zadeh [6] to classical mathematics. In this aspect, the concept of fuzzy subgroups was defined by Rosenfeld [7] and its structure was investigated. The fuzzy groups have been widely studied in [8–11]. In 1975, Zadeh [12] further introduced the concept of intervalvalued fuzzy subsets where the values of the membership functions are intervals of numbers instead of the numbers alone. By using this new concept, Biswas [9] defined the interval-valued fuzzy subgroups with the same nature of the fuzzy subgroups defined by Rosenfeld [7]. A new type of fuzzy subgroup (viz, (∈, ∈ ∨q)-fuzzy subgroup) was therefore introduced by Bhakat and Das [13–15] by using the combined notions of “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets. In fact, these notions were originally introduced by Pu and Liu in [16]. The (∈, ∈ ∨q)-fuzzy subgroup is an important generalization of the fuzzy subgroups defined by Rosenfeld [7] and their structure was described by Bhakat and Das in [17]. In the literature, the relationships between the fuzzy sets and the algebraic hyperstructures (structures) have been considered by many authors such as Ameri, Corsini, Davvaz, Krasner, Leoreanu, Vougiouklis, Zahedi, Zhan and others, for instance, see [18–41]. We notice that Davvaz [23] applied the concept of fuzzy sets to the theory of algebraic hyperstructures and then he defined the fuzzy Hv -submodules of an Hv -modules. Recently, Davvaz et al. [26] considered the intuitionistic fuzzification of the Hv -submodules in an Hv -module and investigated some properties of such Hv -modules. Moreover, Zhan et al. [38] also considered the isomorphism theorems of hypermodules and investigated the fundamental relation on hypermodules. The concept of interval-valued intuitionistic (S, T )-fuzzy Hv -submodules of Hv -modules is consequently introduced and some interesting properties are obtained (see [39]). In this paper, our aim is to introduce the concept of quasi-coincidence of a fuzzy intervalvalue with an interval-valued fuzzy set which generalizes the concept of quasi-coincidence of a fuzzy point in a fuzzy set. By using this new idea, we define the interval-valued (α, β)fuzzy sub-hypermodules of hypermodules. Thus, this is a natural generalization of the fuzzy sub-hypermodules. We shall explore some of the interesting properties of interval-valued (α, β)fuzzy sub-hypermodules. Moreover, some characterization theorems of such hypermodules will be given. Finally, we consider the implication-based interval-valued fuzzy sub-hypermodules of hypermodules. 2

Preliminaries

A hyperstructure is a non-empty set H together with a mapping “ ◦ ” : H × H → P ∗ (H), where P ∗ (H) is the set of all the non-empty subsets of H. If x ∈ H and A, B ∈ P ∗ (H), then by A ◦ B, A ◦ x and x ◦ B, we have A ◦ B = ∪a∈A,b∈B a ◦ b, A ◦ x = A ◦ {x} and x ◦ B = {x} ◦ B, respectively. Now, we call a hyperstructure (H, ◦) a canonical hypergroup [31] if the following axioms are satisfied: (i) For every x, y, z ∈ H, x ◦ (y ◦ z) = (x ◦ y) ◦ z; (ii) For every x, y ∈ H, x ◦ y = y ◦ x; (iii) There exists a 0 ∈ H such that 0 ◦ x = x, for all x ∈ H; (iv) For every x ∈ H, there exists a unique element x ∈ H such that 0 ∈ x ◦ x (we call the element x the opposite of x). Definition 2.1 [9] A hyperring is an algebraic structure (R, +, ·) which satisfies the following axioms : (1) (R, +) is a canonical hypergroup; (we shall write −x for x ); (2) (R, ·) is a semigroup having zero as a bilaterally absorbing element; (3) The multiplication is distributive with respect to the hyperoperation “ + ”. Let (R, +, ·) be a hyperring and A a non-empty subset of R. Then A is called a sub-hyperring of R if (A, +, ·) itself is a hyperring.

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Definition 2.2 [18] A non-empty set M is called a left hypermodule over a hyperring R (R-hypermodule) if (M, +) is a canonical hypergroup and there exists the map “ · ” : R × M → P ∗ (M ) by (r, m) → r.m such that, for all r1 , r2 ∈ R and m1 , m2 ∈ M , we have (i) r1 (m1 + m2 ) = r1 m1 + r1 m2 ; (ii) (r1 + r2 )m1 = r1 m1 + r2 m1 ; (iii) (r1 r2 )m1 = r1 (r2 m1 ). Example 2.3 Let M be an R-module over a unitary ring R and A a subgroup of the multiplicative semigroup of R satisfying the condition aAbA = abA, for every a, b ∈ R. Note that this condition is equivalent to the normality of A only if R\{0} is a group which appears in the case of division rings. Now, we introduce in M an equivalence relation “ ∼ ” defined as follows: x ∼ y ⇐⇒ x = ty, t ∈ A. Let M be the set of the equivalence classes of M modulo ∼. Then, a hyperoperation “ ⊕ ” can be endowed in M by x ⊕ y = {w ∈ M | w ⊆ x + y}, i.e., x ⊕ y consists of all the classes w ∈ M which are contained in the set-wise sum of x and y. Thus, (M , +) becomes a canonical hypergroup. Now, we suppose that R is the quotient hyperring of R by A. Consider an external composition from R × M to M defined by a x = ax for every a ∈ R, x ∈ M . Then the above composition satisfies the axioms of the hypermodule and so M becomes a hypermodule over the ring R. It was proved by Massouros [30] that this hypermodule is strongly related with the analytic projective geometries as well as with the Euclidean spherical geometries. In what follows, all hypermodules considered are left hypermodules. We now call a non-empty subset A of a hypermodule M the sub-hypermodule if (A, +, ·) is a hypermodule. Definition 2.4 A fuzzy subset F of a hypermodule M over a hyperring R is a fuzzy subhypermodule of M if (I1) min{F (x), F (y)} ≤ inf z∈x+y F (z) for all x, y ∈ M ; (II1) F (x) ≤ F (−x) for all x ∈ M ; (III1) F (x) ≤ F (rx) for all r ∈ R and x ∈ M . If F is a fuzzy sub-hypermodule of M , clearly we have F (−x) = F (x), min{F (x), F (y)} ≤ inf F (z), z∈x−y

for all x, y ∈ M . Let M be an R-hypermodule. Then, for a fuzzy subset F of M , the level subset U (F ; t) and the strong level subset U (F ; t> ) are defined by U (F ; t) = {x ∈ M | F (x) ≥ t}, t ∈ [0, 1] and U (F ; t> ) = {x ∈ M | F (x) > t}, t ∈ [0, 1]. The fuzzy sub-hypermodule can be easily characterized in terms of its level subsets and strong level subsets. The following theorem is a direct consequence of the definitions: Theorem 2.5 Let F be a fuzzy subset of an R-hypermodule M . Then the following are equivalent : (1) F is a fuzzy sub-hypermodule of M ; (2) Each non-empty strong level subset of F is a subhypermodule of M ; (3) Each non-empty level subset of F is a subhypermodule of M .

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A fuzzy set F of a hypermodule M of the form  t(= 0) if F (y) = 0 if

y = x, y = x,

is said to be a fuzzy point with support x and value t and is denoted by U (x; t). A fuzzy point U (x; t) is said to “belong to” (resp. be quasi-coincident with) a fuzzy set F , written as U (x; t) ∈ F (resp. U (x; t)qF ) if F (x) ≥ t (resp. F (x) + t > 1). If U (x; t) ∈ F or (resp. and) U (x; t)qF , then we write U (x; t) ∈ ∨q (resp. ∈ ∧q) F . The symbol ∈ ∨q means that ∈ ∨q does not hold. Using the notion of “ belongingness (∈)” and “quasi-coincidence (q)” of fuzzy points with fuzzy subsets, we obtain the concept of an (α, β)-fuzzy subsemigroup, where α and β are any two of {∈, q, ∈ ∨q, ∈ ∧q} with α =∈ ∧q (see [14]). It is noteworthy that the most viable generalization of Rosenfeld’s fuzzy subgroup is the notion of (∈, ∈ ∨q)-fuzzy subgroup. The detailed study of (∈, ∈ ∨q)-fuzzy subgroup has been considered in [42–43]. By an interval number a ˜ we mean (see [12]) an interval [a− , a+ ], where 0 ≤ a− ≤ a+ ≤ 1. The set of all interval numbers is denoted by D[0, 1]. The interval [a, a] is identified with the number a ∈ [0, 1]. + For interval numbers  ai = [a− i , ai ] ∈ D[0, 1], i ∈ I, we define r max{ ai , bi } = [max(a− , b− ), max(a+ , b+ )], i

i

i

i

− + + r min{ ai , bi } = [min(a− i , bi ), min(ai , bi )], + + r inf  ai = [∧i∈I a− r sup  ai = [∨i∈I a− i , ∧i∈I ai ], i , ∨i∈I ai ],

and put − + + a2 ⇐⇒ a− (1)  a1 ≤  1 ≤ a2 and a1 ≤ a2 ; − − + a2 ⇐⇒ a1 = a2 and a1 = a+ (2)  a1 =  2; a2 ⇐⇒  a1 ≤  a2 and  a1 =  a2 ; (3)  a1 <  (4) k a = [ka− , ka+ ], whenever 0 ≤ k ≤ 1. It is clear that (D[0, 1], ≤, ∨, ∧) is a complete lattice with 0 = [0, 0] as its least element and 1 = [1, 1] as its greatest element. By an interval-valued fuzzy set F on X, we mean (see [12]) the set + F = {(x, [μ− F (x), μF (x)]) | x ∈ X}, − + + where μF and μF are two fuzzy subsets of X such that μ− F (x) ≤ μF (x) for all x ∈ X. Putting − +   μ F (x) = [μF (x), μF (x)], we see that F = {(x, μ F (x)) | x ∈ X}, where μ F : X → D[0, 1]. 3

Interval-Valued (α, β)-fuzzy Sub-hypermodules

Basing on the contents in [13–15, 42–43], we can extend the concept of quasi-coincidence of a fuzzy point in a fuzzy set to the concept of quasi-coincidence of a fuzzy interval value within an interval-valued fuzzy set as follows. An interval-valued fuzzy set F of a hypermodule M of the form   t(=[0,0]) if y = x, μ F (y) = [0,0] if y = x, is said to be a fuzzy interval value with support x and interval value  t, and is denoted by U (x;  t). A fuzzy interval value U (x;  t) is said to belong to (resp. be quasi-coincident with) t (resp. an interval-valued fuzzy set F , written as U (x;  t) ∈ F (resp. U (x;  t)qF ) if μ F (x) ≥      (x) + t > [1, 1]). If U (x; t ) ∈ F or (resp. and) U (x; t )qF , then we write U (x; t ) ∈ ∨q (resp. μ F ∈ ∧q) F . The symbol “∈ ∨q” means “∈ ∨q” does not hold. In what follows, let M be a hypermodule over a hyperring R. We now use α and β to denote any one of “ ∈, q, ∈ ∨q” or “ ∈ ∧q” unless otherwise specified. Also, we emphasis − + − + μ F (x) = [μF (x), μF (x)] must satisfy the following properties: [μF (x), μF (x)] < [0.5, 0.5], or − + [0.5, 0.5] ≤ [μF (x), μF (x)] for all x ∈ M .

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Definition 3.1 An interval-valued fuzzy set F of M is called an interval-valued (α, β)-fuzzy sub-hypermodule of M if, for all t, r ∈ (0, 1], a ∈ R and x, y ∈ M , (I2) U (x;  t)αF and U (y; r)αF imply U (z; r min{ t, r})βF , for all z ∈ x + y; (II2) U (x;  t)αF implies U (−x;  t)βF ; (III2) U (x;  t)αF implies U (ax;  t)βF . Let F be an interval-valued fuzzy set of M such that μ F (x) ≤ [0.5, 0.5] for all x ∈ M . t and μ t> Suppose that x ∈ M and t ∈ (0, 1] such that U (x;  t) ∈ ∧qF . Then μ F (x) ≥  F (x) +  t ≤ μ   [1, 1]. It hence follows that [1, 1] < μ F (x) +  F (x) + μ F (x) = 2μ F (x), which implies that μ t)|U (x;  t) ∈ ∧qF } = ∅. Therefore, the case α =∈ ∧q in F (x) > [0.5, 0.5]. This leads to {U (x;  Definition 3.1 can be omitted. Proposition 3.2 Every interval-valued (∈ ∨q, ∈ ∨q)-fuzzy sub-hypermodule of M is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . Proof Let F be an interval-valued (∈ ∨q, ∈ ∨q)-fuzzy sub-hypermodule of M . We need to show only that condition (I2) holds, the other parts of the proof are similar. Let x, y ∈ M and t) ∈ ∨qF and U (y; r) ∈ ∨qF . t, r ∈ (0, 1] be such that U (x;  t) ∈ F and U (y; r) ∈ F . Then U (x;  Since F is an interval-valued (∈ ∨q, ∈ ∨q)-fuzzy sub-hypermodule of M , U (z; r min{ t, r}) ∈ ∨qF , for all z ∈ x + y. This proves condition (I2). The following lemmas are obvious and we omit the details: Lemma 3.3 Every interval-valued (∈, ∈)-fuzzy sub-hypermodule of M is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . Lemma 3.4 If A is a sub-hypermodule of M , then the characteristic function χA of A is an interval-valued (∈, ∈)-fuzzy sub-hypermodule of M . Theorem 3.5 For any subset A of M , χA is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M if and only if A is a sub-hypermodule of M . Proof Let χA be an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . Let x, y ∈ A. Then U (x; [1, 1]) ∈ χA and U (y; [1, 1]) ∈ χA , which imply U (z; [1, 1]) = U (z; r min{[1, 1], [1, 1]}) ∈ ∨q χA , for all z ∈ x + y. Hence χ A (z) > [0, 0] for all z ∈ x + y, and so x + y ∈ A. Let x ∈ A. Then U (x; [1, 1]) ∈ χA , and then U (−x; [1, 1]) ∈ ∨qχA . Hence χ A (−x) > [0, 0], and so −x ∈ A. This proves that A is a sub-hypergroup of (M, +). It is easy to check that A is a sub-hypermodule of M . Conversely, if A is a sub-hypermodule of M , then χA is an interval-valued (∈, ∈)-fuzzy sub-hypermodule of M by Lemma 3.4, and therefore χA is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M by Lemma 3.3. We now give the main result on a general interval-valued (α, β)-fuzzy sub-hypermodule of hypermodules. Theorem 3.6 Let F be a non-zero interval-valued (α, β)-fuzzy sub-hypermodule of M . Then the set U (F ; [0, 0]) = {x ∈ M |μ F (x) > [0, 0]} is a sub-hypermodule of M . Proof Let x, y ∈ U (F ; [0, 0]). Then μ   F (x) > [0, 0] and μ F (y) > [0, 0]. Assume that μ F (z) = (x))αF and U (y; μ  (y))αF , but [0, 0], for all z ∈ x − y. If α ∈ {∈, ∈ ∨q}, then U (x; μ F F (x), μ  (y)})βF for every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, a contradiction. Note that U (z; r min{μ F F U (x; [1, 1])qF and U (y; [1, 1])qF , but, for all z ∈ x−y, U (z; r min{[1, 1], [1, 1]}) = U (z; [1, 1])βF for every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, this is a contradiction. Hence, for all z ∈ x − y, μ F (z) > [0, 0], that is, z ∈ U (F ; [0, 0]), and so x − y ⊆ U (F ; [0, 0]). This proves (U (F ; [0, 0]), +) is a subhypergroup of M . It is easy to check that U (F ; [0, 0]) is a sub-hypermodule of M . Let F be an interval-valued fuzzy set. For every t ∈ [0, 1], the set U (F ;  t) = {x ∈ M |μ F (x)  ≥ t} is called the interval-valued level subset of F .

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An interval-valued fuzzy set F of a hypermodule M is said to be proper if ImF has at least two elements. Two interval-valued fuzzy sets are said to be equivalent if they have same family of interval-valued level subsets. Otherwise, they are said to be non-equivalent. Theorem 3.7 Let M have proper sub-hypermodules. A proper interval-valued (∈, ∈)-fuzzy sub-hypermodule F of M such that card ImF ≥ 3 can be expressed as the union of two proper non-equivalent interval-valued (∈, ∈)-fuzzy sub-hypermodules of M . Proof Let F be a proper interval-valued (∈, ∈)-fuzzy sub-hypermodule of M with ImF = {t0 , t1 , . . . , tn }, where t0 > t1 > · · · > tn and n ≥ 2. Then U (F ; t0 ) ⊆ U (F ; t1 ) ⊆ · · · ⊆ U (F ; tn ) = M is the chain of interval-valued ∈-level sub-hypermodules of F . Define two interval-valued ⎧ fuzzy sets A and B in M by ⎪ r1 , if x ∈ U (F ; t1 ), ⎪ ⎪ ⎪ ⎪ ⎨ t2 , if x ∈ U (F ; t2 )\U (F ; t1 ), μ A (x) = . .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t , if x ∈ U (F ; tn )\U (F ; t n n−1 ), and ⎧ ⎪ t0 , if x ∈ U (F ; t0 ), ⎪ ⎪ ⎪ ⎪ ⎪ if x ∈ U (F ; t1 )\U (F ; t0 ), t1 , ⎪ ⎪ ⎪ ⎪ ⎨ r2 , if x ∈ U (F ; t3 )\U (F ; t1 ), μ B (x) = if x ∈ U (F ; t4 )\U (F ; t3 ), t4 , ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎩ t , if x ∈ U (F ; tn )\U (F ; t n n−1 ), respectively, where t2 < r1 < t1 and t4 < r2 < t2 . Then A and B are interval valued (∈, ∈)-fuzzy sub-hypermodules of M with U (F ; t1 ) ⊆ U (F ; t2 ) ⊆ · · · ⊆ U (F ; tn ) = M and U (F ; t0 ) ⊆ U (F ; t1 ) ⊆ · · · ⊆ U (F ; tn ) = M being respectively chains of interval valued ∈-level sub-hypermodules, and A, B ≤ F . Thus A and B are non-equivalent, and obviously A ∪ B = F . This completes the proof. 4 Interval-valued (∈, ∈ ∨q)-fuzzy Sub-hypermodules In this section, we mainly concentrate on the interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodules of a hypermodule M . We need to extend the fuzzy sub-hypermodules to the interval-valued fuzzy sub-hypermodules of a hypermodule. Definition 4.1 An interval-valued fuzzy set F of M is said to be an interval-valued fuzzy sub-hypermodule of M , if for all a ∈ R and x, y ∈ M , the following inequalities hold :   (I3) r min{μ F (x), μ F (y)} ≤ r inf{μ F (z)|z ∈ x + y}; (x) ≤ μ  (−x); (II3) μ F F  (III3) μ F (x) ≤ μ F (ax). Now, we characterize the interval-valued fuzzy sub-hypermodules by using their level subhypermodules. Theorem 4.2 An interval-valued fuzzy set F of M is an interval-valued fuzzy sub-hypermodule of M if and only if, for any [0, 0] <  t ≤ [1, 1], U (F ;  t)(= ∅) is a sub-hypermodule of M.

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Proof The proof is similar to the proof of Theorem 2.5. We now give the following definition: Definition 4.3 An interval-valued fuzzy set F of M is said to be an interval-valued (∈, ∈ ∨q)fuzzy sub-hypermodule of M if, for all t, r ∈ (0, 1], a ∈ R and x, y ∈ M , (I4) U (x;  t) ∈ F and U (y; r) ∈ F imply U (z; r min{ t, r}) ∈ ∨qF , for all z ∈ x + y; (II4) U (x;  t) ∈ F implies U (−x;  t) ∈ ∨qF ; (III4) U (x;  t) ∈ F implies U (ax;  t) ∈ ∨qF . Note that if F is an interval-valued fuzzy sub-hypermodule of M , according to Definition 4.1, then F is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M , according to Definition 4.3. However, the converse statement is not true in general. Theorem 4.4 Conditions of (I4), (II4) and (III4) in Definition 4.3 are equivalent to the following corresponding conditions, respectively :   (I5) r min{μ F (x), μ F (y), [0.5, 0.5]} ≤ r inf{μ F (z)|z ∈ x + y}, for all x, y ∈ M ; (II5) r min{μ (x), [0.5, 0.5]} ≤ μ  (−x), for all x ∈ M ; F F (III5) r min{μ (x), [0.5, 0.5]} ≤ μ  (ax), for all a ∈ R and x ∈ M . F F Proof (I4)=⇒ (I5): Suppose that x, y ∈ M . Now, we consider the following cases:  (b) r min{μ  (a) r min{μ F (x), μ F (y)} < [0.5, 0.5]; F (x), μ F (y)} ≥ [0.5, 0.5].   Case (a): Assume that there exists z ∈ x + y such that μ F (z) < r min{μ F (x), μ F (y), [0.5, 0.5]}, which implies μ   Choose t such that μ t < r F (z) < r min{μ F (x), μ F (y)}. F (z) <   t) ∈ F and U (y;  t) ∈ F , but U (z;  t)∈ ∨qF , which contradicts min{μ F (x), μ F (y)}. Then U (x;  (I4). Case (b): Assume that μ F (z) < [0.5, 0.5] for some z ∈ x + y. Then U (x; [0.5, 0.5]) ∈ F and U (y; [0.5, 0.5]) ∈ F , but U (z; [0.5, 0.5])∈ ∨qF , a contradiction. Hence (I5) holds. (II4)=⇒ (II5): Suppose that x ∈ M . We now consider the following two cases: (a) μ (b) μ F (x) < [0.5, 0.5]; F (x) ≥ [0.5, 0.5].    < μ Case (a) Assume that μ F (x) = t < [0.5, 0.5] and μ F (−x) = r F (x). Choose s such that r < s <  t and r + s < [1, 1]. Then U (x; s) ∈ F , but U (−x; s)∈ ∨qF , which contradicts (II4). So   μ F (−x) ≥ μ F (x) = r min{μ F (x), [0.5, 0.5]}.   Case (b) Let μ F (x) ≥ [0.5, 0.5]. If μ F (−x) < r min{μ F (x), [0.5, 0.5]}, then U (x; [0.5, 0.5]) ∈ F , but U (−x; [0.5, 0.5])∈ ∨qF , which contradicts (II4). Hence,  μ F (−x) ≥ r min{μ F (x), [0.5, 0.5]}. (III4)=⇒(III5): Suppose that x ∈ M . We now consider the following cases: (a) μ (b) μ F (x) < [0.5, 0.5]; F (x) ≥ [0.5, 0.5].    < μ Case (a) Assume that μ F (x) = t < [0.5, 0.5] and μ F (ax) = r F (x) for some a ∈ R. Choose s such that r < s <  t and r + s < [1, 1]. Then U (x; s) ∈ F , but U (ax; s)∈ ∨qF , which   contradicts (III4). Hence, μ F (ax) ≥ μ F (x) = r min{μ F (x), [0.5, 0.5]}.   Case (b) Let μ F (x) ≥ [0.5, 0.5]. If μ F (ax) < r min{μ F (x), [0.5, 0.5]}, then U (x; [0.5, 0.5]) ∈ F , but U (ax; [0.5, 0.5])∈ ∨qF , which contradicts (III4). Hence, μ  F (ax) ≥ r min{μ F (x), [0.5, 0.5]}.  (I5)=⇒(I4): Let U (x;  t) ∈ F and U (y; r) ∈ F . Then μ (x) ≥ t and μ  (y) ≥ r. For every F F  (z) ≥ r min{ μ  (x), μ  (y), [0.5, 0.5]} ≥ r min{ t , r  , [0.5, 0.5]}. z ∈ x + y, we have μ F F F If r min{ t, r} > [0.5, 0.5], then μ  t, r} > F (z) ≥ [0.5, 0.5]. This implies that μ F (z) + r min{   (z) ≥ r min{ t , r  }. Therefore, U (z; r min{ t , r  }) ∈ ∨qF , [1, 1]. If r min{ t, r} ≤ [0.5, 0.5], then μ F for all z ∈ x + y. t. Now, we have μ  (II5)=⇒(II4): Let U (x;  t) ∈ F . Then μ F (x) ≥  F (−x) ≥ r min{μ F (x), [0.5,  0.5]} ≥ r min{ t, [0.5, 0.5]}, which implies that μ (−x) ≥ t or μ  (−x) ≥ [0.5, 0.5], according to F F  t ≤ [0.5, 0.5] or  t ≥ [0.5, 0.5]. Therefore, U (−x;  t) ∈ ∨qF . (III5)=⇒(III4): The proof is similar to (I5)=⇒(I4).

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By Definition 4.3 and Theorem 4.4, we immediately deduce the following corollary: Corollary 4.5 An interval-valued fuzzy set F of M is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M if and only if the conditions (I5), (II5) and (III5) in Theorem 4.4 hold. Now, we characterize the interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodules by using their level sub-hypermodules. Theorem 4.6 Let F be an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . Then for all [0, 0] <  t ≤ [0.5, 0.5], U (F ;  t) is an empty set or a sub-hypermodule of M . Conversely, if F is an interval-valued fuzzy set of M such that U (F ;  t)(= ∅) is a hyperideal of M for all [0, 0] <  t ≤ [0.5, 0.5], then F is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . Proof Let F be an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M and [0, 0] <  t ≤ t and μ t. Hence, we have [0.5, 0.5]. Let x, y ∈ U (F ;  t). Then μ F (x) ≥  F (x) ≥    t, [0.5, 0.5]} =  t, r inf{μ F (z)|z ∈ x + y} ≥ r min{μ F (x), μ F (y), [0.5, 0.5]} ≥ r min{   μ (−x) ≥ r min{ μ  (x), [0.5, 0.5]} = r min{ t , [0.5, 0.5]} = t , F F and so x + y ⊆ U (F ;  t) and −x ∈ M . Consequently, (U (F ;  t), +) is a sub-hypergroup of (M, +). Also, for every x ∈ U (F ;  t) and a ∈ R, we have  t, [0.5, 0.5]} =  t. μ F (ax) ≥ r min{μ F (x), [0.5, 0.5]} = r min{   This implies that ax ∈ U (F ; t), and therefore U (F ; t) is a sub-hypermodule of M . Conversely, let F be an interval-valued fuzzy set of M such that U (F ;  t)(= ∅) is a subhypermodule of M for all [0, 0] <  t ≤ [0.5, 0.5]. For every x, y ∈ M , we can write     μ F (x) ≥ r min{μ F (−x), [0.5, 0.5]} = k0 ≥ r min{μ F (x), μ F (y), [0.5, 0.5]} = t0 , μ   F (y) ≥ r min{μ F (x), μ F (y), [0.5, 0.5]} = t0 .  Then x, y ∈ U (F ; t0 ) and x ∈ U (F ; k0 ), and so x + y ⊆ U (F ; t0 ) and −x ∈ U (F ; k0 ). Therefore,   r inf{μ F (z)|z ∈ x + y} ≥ r min{μ F (x), μ F (y), [0.5, 0.5]} and μ  F (−x) ≥ r min{μ F (x), [0.5, 0.5]}. (x) ≥ r min{ μ  Also, we have μ F F (x), [0.5, 0.5]} = s0 . Hence x ∈ U (F ; s0 ), which implies  that ax ∈ U (F ; s0 ), and for all a ∈ R, μ F (ax) ≥ r min{μ F (x), [0.5, 0.5]}. Therefore, F is indeed an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M . A corresponding result can be naturally obtained when U (F ;  t) is a sub-hypermodule of M , for all [0.5, 0.5] <  t ≤ [1, 1]. Theorem 4.7 Let F be an interval-valued fuzzy set of M . Then U (F ;  t)(= ∅) is a subhypermodule of M for all [0.5, 0.5] <  t ≤ [1, 1] if and only if   (I6) r min{μ F (x), μ F (y)} ≤ r inf{r max{μ F (z), [0.5, 0.5]}|z ∈ x + y}, for all x, y ∈ M ;  (II6) μ F (x) ≤ r max{μ F (−x), [0.5, 0.5]}, for all x ∈ M ;  (III6) μ F (x) ≤ r max{μ F (ax), [0.5, 0.5]}, for all a ∈ R and x ∈ M .  Proof Assume that U (F ; t) is a sub-hypermodule of M . (I6): If there exist x, y, z ∈ M with z ∈ x + y such that r max{μ   t, F (z), [0.5, 0.5]} < r min{μ F (x), μ F (y)} =     t) and U (F ;  t) is a then [0.5, 0.5] < t ≤ [1, 1], μ F (z) < t, x, y ∈ U (F ; t). Since x, y ∈ U (F ;    sub-hypermodule, so x + y ⊆ U (F ; t) and μ F (z) ≥ t for all z ∈ x + y. However, this contradicts t. Hence, r min{μ   μ F (z) <  F (x), μ F (y)} ≤ r max{μ F (z), [0.5, 0.5]}, for all x, y, z ∈ M with z ∈ x + y. This implies that   r min{μ F (x), μ F (y)} ≤ r inf{r max{μ F (y), [0.5, 0.5]}|z ∈ x + y}, for all x, y ∈ M . Hence (I6) holds.  t. Then (II6): Now, assume that, for some x ∈ M, r max{μ F (−x), [0.5, 0.5]} ≤ μ F (x) =      [0.5, 0.5] < t ≤ [1, 1], μ t) F (−x) < t and x ∈ U (F ; t). Since x ∈ U (F ; t), we obtain −x ∈ U (F ;  t, which is a contradiction. Hence (II6) holds. or μ F (−x) ≥ 

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(III6): The proofs are similar to the proof of (I6). Conversely, suppose that the conditions (I6), (II6) and (III6) hold. We need to show that U (F ;  t) is a sub-hypermodule of M . Assume that [0.5, 0.5] <  t ≤ [1, 1]] and x, y ∈ U (F ;  t). Then (1) [0.5, 0.5] <  t ≤ r min{μ   F (x), μ F (y)} ≤ r inf{r max{μ F (z), [0.5, 0.5]}|z ∈ x + y} < r inf{μ F (z)|z ∈ x + y}; (2) [0.5, 0.5] <  t ≤ μ   F (x) ≤ r max{μ F (−x), [0.5, 0.5]} ≤ μ F (−x); and so x + y ⊆ U (F ;  t), −x ∈ U (F ;  t). Hence (U (F ;  t), +) is a sub-hypergroup of (M, +). Also, we have [0.5, 0.5] <  t ≤ μ   F (x) ≤ r max{μ F (ax), [0.5, 0.5]} < μ F (ax), for all a ∈ R and x ∈ M . This implies that ax ⊆ U (F ;  t). Therefore U (F ;  t) is a sub-hypermodule of M. Let F be an interval-valued fuzzy set of a hypermodule M and J = {α|α ∈ (0, 1] and U (F ; α ) is an empty set or a sub-hypermodule of M }. In particular, if J = (0, 1], then F is an ordinary interval-valued fuzzy sub-hypermodule of M (Theorem 4.2); if J = (0, 0.5), F is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule of M (Theorem 4.6). In [17], Yuan, Zhang and Ren stated the definition of a fuzzy subgroup with thresholds, which is a generalized concept of Rosenfeld’s fuzzy subgroup, and also Bhkat and Das’s fuzzy subgroup. Basing on [17], we can extend the concept of a fuzzy subgroup with thresholds to the concept of interval-valued fuzzy sub-hypermodules with thresholds in the following way: Definition 4.8 Let s, t ∈ [0, 1] and s <  t. Then an interval-valued fuzzy set F of M is called an interval-valued fuzzy sub-hypermodule with thresholds ( s,  t) of M if, for all a ∈ R and x, y ∈ M , (I7) r min{μ  t} ≤ r inf{r max{μ }|z ∈ x + y}; F (x), μ F (y),  F (z), s  (II7) r min{μ (x), t } ≤ r max{ μ  (−x), s  }; F F t} ≤ r max{μ s}. (III7) r min{μ F (x),  F (ax),  Remark If F is an interval-valued fuzzy sub-hypermodule with thresholds of M , then we can conclude that F is an ordinary interval-valued fuzzy sub-hypermodule when s = [0, 0],  t = [1, 1]; and F is an interval-valued (∈, ∈ ∨q)-fuzzy sub-hypermodule when s = [0, 0],  t = [0.5, 0.5]. Now, we characterize the interval-valued fuzzy sub-hypermodules with thresholds by using their level sub-hypermodules. Theorem 4.9 An interval-valued fuzzy set F of M is an interval-valued fuzzy sub-hypermodule with thresholds ( s,  t) of M if and only if U (F ; α )(= ∅) is a sub-hypermodule of M for all s < α ≤ t. Proof Let F be an interval-valued fuzzy sub-hypermodule with thresholds ( s,  t) of M and   and μ . Now s < α  ≤ t. Let x, y ∈ U (F ; α ). Then μ F (x) ≥ α F (y) ≥ α }|z ∈ x + y} ≥ r min{μ  t} r inf{r max{μ F (z), s F (x), μ F (y),  ≥ r min{ α,  t} ≥ α  > s. } > α  > s. This implies that μ Thus, for every z ∈ x + y, we have r max{μ F (z), s F (z) > α , and hence z ∈ U (F ; α ). Consequently, x + y ⊆ U (F ; α ). Now, let x ∈ U (F ;  t). Then } ≥ r min{μ  > s. Therefore, we have proved that μ  r max{μ F (−x), s F (x), t} ≥ α F (−x) ≥ α and −x ∈ U (F ; α ). Also, we have the following inequalities: r max{μ } ≥ r min{μ t} ≥ r min{ α,  t} ≥ α  > s, and hence r max{μ s} ≥ F (ax), s F (x),  F (ax),  (ax) ≥ α  and ax ∈ U (F ; α  ). α  > s. This leads to μ F Therefore, U (F ; α ) is a sub-hypermodule of M for all s < α ≤ t. To prove the converse, we first let F be an interval-valued fuzzy set of M such that U (F ; α )(= ∅) is a sub-hypermodule of R for all s < α ≤ t. If there exist x, y, z ∈ M with z ∈ x+y such that r max{μ } < r min{μ  t} = α , then s < α ≤ t], μ  and x, y ∈ U (F ; α ). F (z), s F (x), μ F (y),  F (z) < α Since U (F ; α ) is a sub-hypermodule of M and x, y ∈ U (F ; α ), we have, x + y ⊆ U (F ; α ) and

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thereby, μ , for all z ∈ x + y. This clearly contradicts to μ . Therefore, F (z) ≥ α F (z) < α  (x), μ  (y), t } ≤ r max{ μ  (z), s  }, for all x, y ∈ M with z ∈ x + y. r min{μ F F F Now, we assume that there exists x0 ∈ M such that r min{μ t} > α  = r max F (x0 ),   {μ (−x ), s  }. Then x ∈ U (F ; α  ), s  < α  ≤ t and μ  (−x ) < α  . Since U (F ; α ) is a F 0 0 F 0 (−x ) ≥ α  . This clearly contradicts μ  (−x ) < α  . Therefore sub-hypermodule of M ,μ F 0 F 0 t} ≤ r max{μ } for any x ∈ M . Also, if there exist a ∈ R and x ∈ M such r min{μ F (x),  F (−x), s  then s < β ≤    } < r min{μ  t} = β, t, μ that r max{μ F (ax), s F (x), μ F (y),  F (ax) < β, x ∈ U (F ; β),    and so ax ∈ U (F ; β). Hence μ  F (xy) ≥ β. This again contradicts μ F (ax) < β. Hence, t} ≤ r max{μ }, for all a ∈ R and x ∈ M . This shows that F is indeed an r min{μ F (x),  F (ax), s interval-valued fuzzy sub-hypermodule with thresholds ( s,  t) of M . 5

Implication-based Interval-valued Fuzzy Sub-hypermodules

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 ∧, ∨, ¬ and → in fuzzy logic can also be defined by using the truth tables, and the extension principle can be applied to derive definitions of the operators. Of course, if various implication operators have been defined, then we need to show only a selection of them in the following table. We now use α to denote the degree of truth (or degree of membership) of the premise, β to denote the respective values for the consequence, and I to denote the resulting degree of truth for the implication. The table is given below: Name

Definition of Implication Operators

Early Zadeh

Im (α, β) = max{1 − α, min{α, β}}

Lukasiewicz

Ia (α, β) = min{1, 1 − α + β}  1, if α ≤ β, Ig (α, β) = β, if α > β  1, if α ≤ β, Icg (α, β) = 1 − α, if α > β  1, if α ≤ β, Igr (α, β) = 0, if α > β

Standard Star (Godel) Contraposition of Godel Gaines-Rescher Kleene-Dienes

Ib (α, β) = max{1 − α, β}

We remark here that the “quality” of these implication operators can be evaluated either empirically or axiomatically. In the following definition, we consider the implication operators in the Lukasiewicz system of continuous-valued logic. Definition 5.1 An interval-valued fuzzy set F of M is called an interval-valued fuzzifying sub-hypermodule of M if it satisfies the following conditions : (I8) For any x, y ∈ M, |= [r min{[x ∈ F ], [y ∈ F ]} → [∀z ∈ x + y, z ∈ F ]]; (II8) For any x ∈ M, |= [[x ∈ F ] → [−x ∈ F ]]; (III8) For any a ∈ R and x ∈ M, |= [r min{[x ∈ F ], [y ∈ F ]} → [ax ∈ F ]]. Clearly, Definition 5.1 is equivalent to Definition 4.1. Therefore the interval-valued fuzzifying sub-hypermodule is an ordinary interval-valued fuzzy sub-hypermodule. We now introduce the concept of interval-valued t-tautology, that is, |=t P if and only if [P ] ≥  t for all valuations. Basing on [17], we can extend the concept of implication-based fuzzy sub-hypermodules in the following way:

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Definition 5.2 Let F be an interval-valued fuzzy set of M and t ∈ (0, 1] be a fixed number. Then F is called a t-implication-based interval-valued fuzzy sub-hypermodule of M if it satisfies : (I9) For any x, y ∈ M, |=t [r min{[x ∈ F ], [y ∈ F ]} → [∀z ∈ x + y, z ∈ F ]]; (II9) For any x ∈ M, |=t [[x ∈ F ] → [−x ∈ F ]]; (III9) For any a ∈ R and x ∈ M, |=t [r min{[x ∈ F ], [y ∈ F ]} → [xy ∈ F ]]. Now, let I be an implication operator. Then we have Corollary 5.3 An interval-valued fuzzy set F of M is a t-implication-based interval-valued fuzzy sub-hypermodule of M if and only if all the following statements hold : (I10) I(r max{μ   t, for all x, y ∈ M ; F (x), μ F (y)}, r inf{μ F (z)|z ∈ x + y}) ≥  (II10) For any x ∈ M, I(r min{μ  t; F (x), μ F (−x)}) ≥  (III10) I(μ  t, for all a ∈ R and x ∈ M . F (x), μ F (ax)) ≥  Let F be an interval-valued fuzzy set of M . Then we have the following results: Theorem 5.4 (i) Let I = Igr . Then F is a 0.5-implication-based interval-valued fuzzy subhypermodule of M if and only if F is an interval-valued fuzzy sub-hypermodule with thresholds ( r = [0, 0], s = [1, 1]) of M ; (ii) Let I = Ig . Then F is a 0.5-implication-based interval-valued fuzzy sub-hypermodule of M if and only if F is an interval-valued fuzzy sub-hypermodule with thresholds ( r = [0, 0], s = [0.5, 0.5]) of M ; (iii) Let I = Icg . Then F is a 0.5-implication-based interval-valued fuzzy sub-hypermodule of M if and only if F is an interval-valued fuzzy sub-hypermodule with thresholds ( r = [0.5, 0.5], s = [1, 1]) of M . Proof The proof is straightforward, by considering the definitions. 6

Conclusions

The aim of this paper is to introduce a new kind of interval-valued fuzzy sub-hypermodules of hypermodules and to investigate their related properties. Also, we consider the definition of implication operators in the Lukusiewicz system of continuous-valued logic for interval-valued fuzzy sub-hypermodules. Our results can be applied to other algebraic hyperstructures. In fact, by considering the notion of an interval-valued (α, β)-fuzzy sub-hypermodule, we can deduce twelve different types of such structures, resulting from three choices of α and four choices of β. However, in this paper, we mainly discuss the (∈, ∈)-type, the (∈ ∨q, ∈ ∨q)-type and the (∈, ∈ ∨q)-type. In our future research, we will consider other types of fuzzy sub-hyperstructure with relations among them. Also, we will consider the applications of such fuzzy sub-hyperstructures in information sciences and general systems. Acknowledgements The authors are highly grateful to the referees for their valuable comments and suggestions for improving the paper. References [1] Marty, F.: Sur une generalization de la notation de groupe, 8th Congress Math. Scandianaves, Stockholm, 45–49, 1934 [2] Corsini, P.: Prolegomena of hypergroup theory, Aviani, editor, 1993 [3] Davvaz, B.: A brief survey of the theory of Hv -structures, Proc. 8th Int. Congress AHA, Greece Spanids Press, 39–70, 2003 [4] Vougiouklis, T.: Hyperstructures and their representations, Hadronic Press Inc., Palm Harber, USA, 1994 [5] Corsini, P., Leoreanu, V.: Applications of hyperstructure theory, Advances in Mathematics, Kluwer Academic Publishers, Dordrecht, 2003 [6] Zadeh, L. A.: Fuzzy sets. Inform. Control, 8, 338–353 (1965) [7] Rosenfeld, A.: Fuzzy groups. J. Math. Anal. Appl., 35, (1971) 512-517. [8] Bhattacharya, P.: Fuzzy subgroups: Some charcterizations(II). Fuzzy Sets Syst., 38, 293–297 (1986)

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