Fuzzy Γ-Hyperideals in Γ-Hypersemirings by Using Triangular Norms

The concept of Γ-semihyperrings was introduced by Dehkordi and Davvaz as a generalization of semirings, semihyperrings, and Γ-semiring. In this paper, by using the notion of triangular norms, we define the concept of triangular fuzzy sub-Γ-semihyperrings as well as triangular fuzzy Γ-hyperideals of a Γ-semihyperring, and we study a few results in this respect.


Introduction
In [1], Nobusawa introduced Γ-rings as a generalization of ternary rings. Let be an additive group whose elements are denoted by , , , . . . and Γ another additive group whose elements are , , , . . .. Suppose that is defined to be an element of and that is defined to be an element of Γ for every , , , and . If the products satisfy the following three conditions: (1) ( 1 + 2 ) = 1 + 2 , ( 1 + 2 ) = 1 + 2 , ( 1 + 2 ) = 1 + 2 ; (2) ( ) = ( ) = ( ) ; (3) if = 0 for any and in , then = 0; then is called a Γ-ring in the sense of Nobusawa [1]. Barnes [2] weakened slightly the conditions in the definition of Γring and gave a new definition of a Γ-ring. Let and Γ be two additive abelian groups. Suppose that there is a mapping from × Γ × → (sending ( , , ) into such that (1) ( 1 + 2 ) = 1 + 2 , ( 1 + 2 ) = 1 + 2 , ( 1 + 2 ) = 1 + 2 ; (2) ( ) = ( ); then is called a Γ-ring in the sense of Barnes [2]. Nowadays, Γ-rings mean the Γ-rings due to Barnes and other Γ-rings are known as Γ -rings, that is, gamma rings in the sense of Nobusawa. Barnes [2], Luh [3], and Kyuno [4] studied the structure of Γ-rings and obtained various generalization analogous to corresponding parts in ring theory. The notion of Γ-semirings was introduced by Rao [5] as a generalization of semirings as well as Γ-rings. Subsequently, by introducing the notion of operator semirings of a Γ-semiring, Dutta and Sardar [6] enriched the theory of Γ-semirings. Algebraic hyperstructures represent a natural extension of classical algebraic structures and they were introduced by the French mathematician Marty [7]. Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic structure, the composition of two elements is a set. Since then, hundreds of papers and several books have been written on this topic, for example, see [8][9][10]. In [11,12], Dehkordi and Davvaz studied the notion of a Γ-semihyperring as a generalization of semiring, semihyperring, and Γ-semiring.
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Zadeh (1965) as an extension of the classical notion of sets [13]. Let be a set. A fuzzy subset of is characterized by a membership function : → [0, 1] which associates with each point ∈ its grade or degree of membership ( ) ∈ [0, 1]. Fuzzy sets generalize classical sets since the characteristic functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. After the introduction of fuzzy sets by Zadeh, reconsideration of the concept of classical mathematics began. In 1971, Rosenfeld [14] introduced fuzzy sets in the context of group theory and formulated 2 The Scientific World Journal the concept of a fuzzy subgroup of a group. Das characterized fuzzy subgroups by their level of subgroups in [15], since then many notions of fuzzy group theory can be equivalently characterized with the help of notion of level subgroups. The concept of a fuzzy ideal of a ring was introduced by Liu [16]. In 1992, Jun and Lee [17] introduced the notion of fuzzy ideals in Γ-rings and studied a few properties. In [6], Dutta and Sardar studied the structures of fuzzy ideals of Γ-rings. Also, see [18]. The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures. In [19], Davvaz introduced the concept of fuzzy V -ideals of V -rings. Then, this concept was studied in depth in several papers, for example, see [20]. Also, see [21]. In [22,23], Ersoy and Davvaz investigated some properties of fuzzy Γhyperideals of Γ-semihyperring. Now, in this paper, we define the concept of triangular fuzzy sub-Γ-semihyperrings and fuzzy Γ-hyperideals of a Γ-semihyperring by using triangular norms, and we study a few results in this respect.
A right (left) Γ-hyperideal of a Γ-semihyperring is an additive sub-semihypergroup ( , +) such that Γ ⊆ ( Γ ⊆ ). If is both right and left Γ-hyperideal of , then we say that is a two-sided Γ-hyperideal or simply a Γhyperideal of . In [22,23], Ersoy and Davvaz studied fuzzy Γ-hyperideals of Γ-semihyperrings. We recall the notion of a fuzzy Γ-hyperideal of a Γ-semihyperring. Let be a Γsemihyperring and be a fuzzy subset of .
The concept of a triangular norm was introduced by Menger [26] in order to generalize the triangular inequality of a metric. The current notion of a t-norm and its dual operation is due to Schweizer and Sklar [27]. By a t-norm we mean a function : From an axiomatic point of view, t-norms and t-conorms differ only with respect to their respective boundary conditions. In fact, the concepts of t-norms and t-conorms are dual in some sense. Anthony and Sherwood [28] redefined a fuzzy subgroup of a group by using the notion of t-norm.

-Fuzzy Sub-Γ-Semihyperrings and -Fuzzy Γ-Hyperideals
In this section, we define the notion of -fuzzy sub-Γ-semihyperrings and -fuzzy Γ-hyperideals of a Γsemihyperring and we study some of their properties. Let be a t-norm and be a fuzzy subset of a Γ-semihyperring . Then, we say has imaginable property if Im ⊆ Δ .

Definition 1.
Let be a Γ-semihyperring, be a -norm, and be a fuzzy subset of . Then, is called a -fuzzy sub-Γsemihyperring of if for all , ∈ and for all ∈ Γ.
A -fuzzy sub-Γ-semihyperring of is said to be imaginable if it satisfies the imaginable property. Clearly, if is a Γ-semiring, then is a -fuzzy sub-Γ-semiring of when for all , ∈ and for all ∈ Γ.

(5)
Proof. The proof is straightforward by mathematical induction.

Lemma 4.
Let be a Γ-semihyperring, be a -norm, and be a -fuzzy sub-Γ-semihyperring of . Let and be nonempty subsets of . Then for all ∈ Γ.
Proof. The proof is straightforward.

Theorem 5.
Let be a Γ-semihyperring, be a -norm, and be a fuzzy subset of with imaginable property and the maximum of Im . Then, the following two statements are equivalent: (1) is a -fuzzy sub-Γ-semihyperring of , and so ( ( ), ( )) ∈ Δ . Assume that = ( ( ), ( )). If = 0, then (2) is called a -fuzzy right Γ-hyperideal of if (3) is called a -fuzzy Γ-hyperideal of if it is both a -fuzzy left Γ-hyperideal and a -fuzzy right Γhyperideal of .

Theorem 7.
Let be a Γ-semihyperring, be a -norm, and be a fuzzy subset of with imaginable property and the maximum of Im . Then, the following two statements are equivalent: (1) is a -fuzzy Γ-hyperideal of , is a Γ-hyperideal of whenever ∈ Δ and 0 < ≤ .
Proof. The proof is similar to the proof of Theorem 5.

(1) is a Min-fuzzy sub-Γ-semihyperring of if and only if every nonempty level subset is a sub-Γ-semihyperring of ;
(2) is a Min-fuzzy Γ-hyperideal of if and only if every nonempty level subset is a Γ-hyperideal of .

(1) the characteristic function is a -fuzzy sub-Γsemihyperring of if and only if is a sub-Γsemihyperring of ;
(2) the characteristic function is a -fuzzy Γ-hyperideal of if and only if is a Γ-hyperidealof .
Proof. The proof is similar to the proof of Theorem 2.6 in [29].
Definition 11. Let 1 and 2 be Γ 1 and Γ 2 -semihyperrings, respectively. If there exists a map : 1 → 2 and a bijection : Γ 1 → Γ 2 such that for all , ∈ 1 and ∈ Γ, then we say ( , ) is a homomorphism from 1 to 2 . Also, if is a bijection then ( , ) is called an isomorphism and 1 and 2 are isomorphic.
If is continuous, then is infinitely distributive [30].
In [12], Dehkordi and Davvaz studied Noetherian and Artinian Γ-semihyperrings in crisp case. A collection A of subsets of a Γ-semihyperring satisfies the ascending chain condition (or Acc) if there does not exist a properly ascending infinite chain 1 ⊂ 2 ⊂ ⋅ ⋅ ⋅ of subsets from A. Recall that a subset ∈ A is a maximal element of A if there does not exist a subset in A that properly contains . Similar to [18], in the following, we obtain some results related to fuzzy sets and Noetherian Γ-semihyperrings.
Definition 20 (see [12]). A Γ-semihyperring is right (left) Noetherian if the equivalent conditions of the above proposition are satisfied. In the same way, we can define an Artinian Γ-semihyperring. Let be a Γ-hyperideal of a Γsemihyperring and be a Noetherian Γ-semihyperring. Then, is called a Noetherian Γ-hyperideal of .

6
The Scientific World Journal Theorem 22. Let { | ∈ N} be a family of Γ-hyperideals of a Γ-semihyperring , where 1 ⊃ 2 ⊃ 3 ⋅ ⋅ ⋅ . Let be a fuzzy subset of defined by for all ∈ , where 0 stands for . Let be a t-norm with Im ⊆ Δ . Then, is a -fuzzy Γ-hyperideal of .

Theorem 23.
Let be a Γ-semiring satisfying descending chain condition, let be a fuzzy subset of , and let be a -norm with Im ⊆ Δ . Let be a -fuzzy Γ-hyperideal of . If a sequence of elements of Im is strictly increasing, then has finite number of values.
For a family { | ∈ Λ} of fuzzy subsets in , we define the join ∨ ∈Λ and the meet ∨ ∈Λ as follows: for all ∈ , where Λ is any index set.