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

Abstract

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.

1. Introduction

In [1], Nobusawa introduced Γ-rings as a generalization of ternary rings. Let 𝑀 be an additive group whose elements are denoted by 𝑎,𝑏,𝑐,… and Γ another additive group whose elements are 𝛾,𝛽,𝛼,…. Suppose that 𝑎𝛾𝑏 is defined to be an element of 𝑀 and that 𝛾𝑎𝛽 is defined to be an element of Γ for every 𝑎,𝑏,𝛾, and 𝛽. If the products satisfy the following three conditions: (1) (𝑎₁+𝑎₂)𝛾𝑏=𝑎₁𝛾𝑏+𝑎₂𝛾𝑏, 𝑎(𝛾₁+𝛾₂)𝑏=𝑎𝛾₁𝑏+𝑎𝛾₂𝑏, 𝑎𝛾(𝑏₁+𝑏₂)=𝑎𝛾𝑏₁+𝑎𝛾𝑏₂; (2) (𝑎𝛾𝑏)𝛽𝑐=𝑎𝛾(𝑏𝛽𝑐)=𝑎(𝛾𝑏𝛽)𝑐; (3) if 𝑎𝛾𝑏=0 for any 𝑎 and 𝑏 in 𝑀, then 𝛾=0; then 𝑀 is called a Γ-ring in the sense of Nobusawa [1]. Barnes [2] weakened slightly the conditions in the definition of Γ-ring and gave a new definition of a Γ-ring. Let 𝑀 and Γ be two additive abelian groups. Suppose that there is a mapping from 𝑀×Γ×𝑀→𝑀 (sending (𝑎,𝛾,𝑏) into 𝑎𝛾𝑏 such that (1)  (𝑎₁+𝑎₂)𝛾𝑏=𝑎₁𝛾𝑏+𝑎₂𝛾𝑏, 𝑎(𝛾₁+𝛾₂)𝑏=𝑎𝛾₁𝑏+𝑎𝛾₂𝑏, 𝑎𝛾(𝑏₁+𝑏₂)=𝑎𝛾𝑏₁+𝑎𝛾𝑏₂; (2)  (𝑎𝛾𝑏)𝛽𝑐=𝑎𝛾(𝑏𝛽𝑐); then 𝑀 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_𝑁-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 [8–10]. 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 𝑋 be a set. A fuzzy subset 𝐴 of 𝑋 is characterized by a membership function 𝜇_𝐴∶𝑋→[0,1] which associates with each point 𝑥∈𝑋 its grade or degree of membership 𝜇_𝜇(𝑥)∈[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 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 𝐻_𝑣-ideals of 𝐻_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 Γ-hyperideals 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.

2. Basic Concepts

Let 𝐻 be a nonempty set and let ℘^∗(𝐻) be the set of all nonempty subsets of 𝐻. A hyperoperation on 𝐻 is a map ∘∶𝐻×𝐻→℘^∗(𝐻) and the couple (𝐻,∘) is called a hypergroupoid. If 𝐴 and 𝐵 are nonempty subsets of 𝐻, then we denote 𝐴∘𝐵=⋃_{𝑎∈𝐴,𝑏∈𝐵}𝑎∘𝑏, 𝑥∘𝐴={𝑥}∘𝐴 and 𝐴∘𝑥=𝐴∘{𝑥}. A hypergroupoid (𝐻,∘) is called a semihypergroup if for all 𝑥,𝑦,𝑧 of 𝐻 we have (𝑥∘𝑦)∘𝑧=𝑥∘(𝑦∘𝑧), which means that ⋃_{𝑢∈𝑥∘𝑦}𝑢∘𝑧=⋃_{𝑣∈𝑦∘𝑧}𝑥∘𝑣. A semihyperring is an algebraic structure (𝑅,+,⋅) which satisfies the following properties: (1)  (𝑅,+) is a commutative semihypergroup; that is, (𝑥+𝑦)+𝑧=𝑥+(𝑦+𝑧) and 𝑥+𝑦=𝑦+𝑥 for all 𝑥,𝑦,𝑧∈𝑅; (2)  (𝑅,⋅) is a semihypergroup; (3) the hyperoperation ⋅ is distributive with respect to the hyperoperation +; that is, 𝑥⋅(𝑦+𝑧)=𝑥⋅𝑦+𝑥⋅𝑧,(𝑥+𝑦)⋅𝑧=𝑥⋅𝑧+𝑦⋅𝑧 for all 𝑥,𝑦,𝑧∈𝑅; (4) the element 0∈𝑅 is an absorbing element; that is 𝑥⋅0=0⋅𝑥=0 for all 𝑥∈𝑅. A semihyperring (𝑅,+,⋅) is called commutative if and only if 𝑎⋅𝑏=𝑏⋅𝑎 for all 𝑎,𝑏∈𝑅. Vougiouklis in [24] studied the notion of semihyperrings in a general form; that is, both the sum and product are hyperoperations; also see [25]. A semihyperring (𝑅,+,⋅) with identity 1_𝑅∈𝑅 means that 1_𝑅⋅𝑥=𝑥⋅1_𝑅=𝑥 for all 𝑥∈𝑅. The concept of Γ-semihyperring is introduced and studied by Dehkordi and Davvaz [11]. Let 𝑅 be a commutative semihypergroup and Γ be a commutative group. Then, 𝑅 is called a Γ-semihyperring if there exists a map 𝑅×Γ×𝑅→℘^∗(𝑅) (image to be denoted by 𝑎𝛼𝑏 for 𝑎,𝑏∈𝑅 and 𝛼∈Γ) and ℘^∗(𝑅) is the set of all nonempty subsets of 𝑅 satisfying the following conditions:(1)𝑎𝛼(𝑏+𝑐)=𝑎𝛼𝑏+𝑎𝛼𝑐, (2)(𝑎+𝑏)𝛼𝑐=𝑎𝛼𝑐+𝑏𝛼𝑐, (3)𝑎(𝛼+𝛽)𝑏=𝑎𝛼𝑏+𝑎𝛽𝑏, (4)𝑎𝛼(𝑏𝛽𝑐)=(𝑎𝛼𝑏)𝛽𝑐, for all 𝑎,𝑏,𝑐∈𝑅 and for all 𝛼,𝛽∈Γ. One can find many examples of Γ-semihyperrings in [11, 12]. In the above definition if 𝑅 is a semigroup, then 𝑅 is called a multiplicative Γ-semihyperring. A Γ-semihyperring 𝑅 is called commutative if 𝑥𝛼𝑦=𝑦𝛼𝑥 for every 𝑥,𝑦∈𝑅 and 𝛼∈Γ. We say that Γ-semihyperring 𝑅 with zero if there exists 0∈𝑅 such that 𝑎∈𝑎+0, 0∈0𝛼𝑎,0∈𝑎𝛼0 for all 𝑎∈𝑅 and 𝛼∈Γ. Let 𝐴 and 𝐵 be two nonempty subsets of Γ-semihyperring 𝑅 and 𝑥∈𝑅. We define 𝐴+𝐵={𝑥∣𝑥∈𝑎+𝑏;𝑎∈𝐴,𝑏∈𝐵} and 𝐴Γ𝐵={𝑥∣𝑥∈𝑎𝛼𝑏;𝑎∈𝐴,𝑏∈𝐵,𝛼∈Γ}. A nonempty subset 𝑅₁ of Γ-semihyperring 𝑅 is called a sub-Γ-semihyperring if it is closed with respect to the multiplication and addition. In other words, a nonempty subset 𝑅₁ of Γ-semihyperring 𝑅 is a sub-Γ-semihypergroup if 𝑅₁+𝑅₁⊆𝑅₁ and 𝑅₁Γ𝑅₁⊆𝑅₁.

A right (left) Γ-hyperideal 𝐼 of a Γ-semihyperring 𝑅 is an additive sub-semihypergroup (𝑅,+) such that 𝐼Γ𝑅⊆𝐼 (𝑅Γ𝐼⊆𝐼). If 𝐼 is both right and left Γ-hyperideal of 𝑅, then we say that 𝐼 is a two-sided Γ-hyperideal or simply a Γ-hyperideal of 𝑅. 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 𝑅 be a Γ-semihyperring and 𝜇 be a fuzzy subset of 𝑅. Then (1)  𝜇 is called a fuzzy left Γ-hyperideal of 𝑅 if min{𝜇(𝑥),𝜇(𝑦)}≤inf_{𝑧∈𝑥+𝑦}{𝜇(𝑧)} and 𝜇(𝑦)≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇(𝑧)} for all 𝑥,𝑦∈𝑅 and 𝛾∈Γ; (2)  𝜇 is called a fuzzy right Γ-hyperideal of 𝑅 if min{𝜇(𝑥),𝜇(𝑦)}≤inf_{𝑧∈𝑥+𝑦}{𝜇(𝑧)} and 𝜇(𝑥)≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇(𝑧)} for all 𝑥,𝑦∈𝑅 and 𝛾∈Γ; (3)  𝜇 is called a fuzzy Γ-hyperideal of 𝑅 if it is both a fuzzy left Γ-hyperideal and a fuzzy right Γ-hyperideal of 𝑅.

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 𝑇 we mean a function 𝑇∶[0,1]×[0,1]→[0,1] satisfying the following conditions: (T1) 𝑇(𝑥,𝑦)=𝑇(𝑦,𝑥); (T2) 𝑇(𝑥,𝑇(𝑦,𝑧))=𝑇(𝑇(𝑥,𝑦),𝑧); (T3) 𝑇(𝑥,𝑦)≤𝑇(𝑥,𝑧) if 𝑦≤𝑧; (T4) 𝑇(𝑥,1)=𝑥, for all 𝑥,𝑦,𝑧∈[0,1]. For every t-norm 𝑇, we set Δ_𝑇={𝑥∈[0,1],𝑇(𝑥,𝑥)=𝑥}. A t-norm on [0,1] is called a continuous t-norm if 𝑇 is a continuous function from [0,1]×[0,1] to [0,1] with respect to the usual topology. Note that the function “Min” is a continuous t-norm. A triangular conorm (t-norm for short) is a binary operation 𝑆 on the unit interval [0,1], that is, a function 𝑆∶[0,1]×[0,1]→[0,1], which for all 𝑥,𝑦,𝑧∈[0,1] satisfies (T1)–(T3) and (S4) 𝑆(𝑥,0)=𝑥. 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.

3. 𝑇-Fuzzy Sub-Γ-Semihyperrings and 𝑇-Fuzzy Γ-Hyperideals

In this section, we define the notion of 𝑇-fuzzy sub-Γ-semihyperrings and 𝑇-fuzzy Γ-hyperideals of a Γ-semihyperring and we study some of their properties. Let 𝑇 be a t-norm and 𝜇 be a fuzzy subset of a Γ-semihyperring 𝑅. Then, we say 𝜇 has imaginable property if Im𝜇⊆Δ_𝑇.

Definition 1. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a fuzzy subset of 𝑅. Then, 𝜇 is called a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅 if(1)𝑇(𝜇(𝑥),𝜇(𝑦))≤inf_{𝑧∈𝑥+𝑦}{𝜇(𝑧)},(2)𝑇(𝜇(𝑥),𝜇(𝑦))≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇(𝑧)}, for all 𝑥,𝑦∈𝑅 and for all 𝛾∈Γ.

A 𝑇-fuzzy sub-Γ-semihyperring of 𝑅 is said to be imaginable if it satisfies the imaginable property. Clearly, if 𝑅 is a Γ-semiring, then 𝜇 is a 𝑇-fuzzy sub-Γ-semiring of 𝑅 when (1′)𝑇(𝜇(𝑥),𝜇(𝑦))≤𝜇(𝑥+𝑦), (2′)𝑇(𝜇(𝑥),𝜇(𝑦))≤𝜇(𝑥𝛾𝑦),for all 𝑥,𝑦∈𝑅 and for all 𝛾∈Γ.

Example 2. Suppose that 𝑅=ℕ, the set of natural numbers, and Γ is a nonempty subset of 𝑅. For any 𝑥,𝑦∈𝑅 and 𝛾∈Γ, we define 𝑥+𝑦={𝑥,𝑦}and 𝑥𝛾𝑦={𝑥,𝛾,𝑦}. Then, 𝑅 is a Γ-semihyperring. We define the fuzzy subset 𝜇 of 𝑅 by 𝜇(𝑥)=34if5𝑥∈Γ9otherwise (1) and we consider the 𝑡-norm 𝑇(𝑟,𝑠)=𝑟𝑠/(2−(𝑟+𝑠−𝑟𝑠)), where 𝑟,𝑠∈[0,1]. Then, for any 𝑥,𝑦∈𝑅 and 𝛾∈Γ, we have inf_{𝑧∈𝑥+𝑦}𝜇(𝑧𝜇(𝑥)𝑦=3)=min,𝜇4if5𝑥,𝑦∈Γ9otherwise,inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧𝜇(𝑥)𝛾𝑦=5)=min,𝜇,𝜇9. (2) On the other hand, we have the following cases:(1)𝑥,𝑦∈Γ, (2)𝑥∉Γ and 𝑦∈Γ(or,𝑥∈Γand𝑦∉Γ),(3)𝑥,𝑦∉Γ. Regarding the above cases, we have 𝑇(3/4,3/4)=9/17, 𝑇(3/4,5/9)=15/40, and 𝑇(5/9,5/9)=25/81. Thus, in every case, we obtain 𝑇𝜇(𝑥),𝜇𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧,𝑇𝜇(𝑥)𝑦),𝜇≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧). (3) Therefore, 𝜇 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅.

Lemma 3. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a 𝑇-fuzzy sub-Γ-semihyperring of 𝑆. Then 𝑇_𝑛𝜇𝑥₁,…,𝜇𝑥_𝑛≤inf_{𝑧∈𝑥1+⋯+𝑥𝑛}𝜇(𝑧,𝑇_𝑛)𝜇𝑥₁,…,𝜇𝑥_𝑛≤inf_{𝑧∈𝑥1𝛾⋯𝛾𝑥𝑛}𝜇(𝑧), (4) for all 𝑥₁,…,𝑥_𝑛∈𝑅 and 𝛾∈Γ, where 𝑇_𝑛𝑡₁,…,𝑡_𝑛=𝑡₁𝑇𝑡𝑖𝑓𝑛=1₁,𝑡₂𝑇𝑡𝑖𝑓𝑛=2_𝑖,𝑇_𝑛−1𝑡₁,…,𝑡_𝑖−1,𝑡_𝑖+1,…,𝑡_𝑛𝑖𝑓𝑛>2. (5)

Proof. The proof is straightforward by mathematical induction.

Lemma 4. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a 𝑇-fuzzy sub-Γ-semihyperring of 𝑆. Let 𝐴 and 𝐵 be nonempty subsets of 𝑅. Then 𝑇inf_𝑎∈𝐴𝜇(𝑎),inf_𝑏∈𝐵𝜇(𝑏)≤inf_{𝑧∈𝐴+𝐵}𝜇(𝑧,𝑇)inf_𝑎∈𝐴𝜇(𝑎),inf_𝑏∈𝐵𝜇(𝑏)≤inf_{𝑧∈𝐴𝛾𝐵}𝜇(𝑧), (6) for all 𝛾∈Γ.

Proof. The proof is straightforward.

Theorem 5. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a fuzzy subset of 𝑆 with imaginable property and 𝑏 the maximum of Im 𝜇. Then, the following two statements are equivalent:(1)𝜇 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑆,(2)𝜇⁻¹[𝑎,𝑏] is a sub-Γ-semihyperring of 𝑆 whenever 𝑎∈Δ_𝑇 and 0<𝑎≤𝑏.

Proof. (1)⇒(2): Suppose that 𝑎∈Δ_𝑇 and 0<𝑎≤𝑏. If 𝑥,𝑦∈𝜇⁻¹[𝑎,𝑏], then inf_{𝑧∈𝑥+𝑦}{𝜇(𝑧)}≥𝑇(𝜇(𝑥),𝜇(𝑦))≥𝑇(𝑎,𝑎)=𝑎, which implies that 𝑥+𝑦⊆𝜇⁻¹[𝑎,𝑏]. Similarly, assume that 𝑎∈Δ_𝑇 and 0<𝑎≤𝑏. If 𝑥,𝑦∈𝜇⁻¹[𝑎,𝑏] and 𝛾∈Γ, then inf_{𝑧∈𝑥𝛾𝑦}{𝜇(𝑧)}≥𝑇(𝜇(𝑥),𝜇(𝑦))≥𝑇(𝑎,𝑎)=𝑎. Then, we have 𝑥𝛾𝑦⊆𝜇⁻¹[𝑎,𝑏], and so 𝜇⁻¹[𝑎,𝑏] is a sub-Γ-semihyperring of 𝑅.
(2)⇒(1): Suppose that 𝑥,𝑦∈𝑆 and 𝛾∈Γ. Since 𝐼𝑚𝜇⊆Δ_𝑇, both 𝜇(𝑥) and 𝜇(𝑦) are in Δ_𝑇. Now, we have 𝑇𝑇𝜇(𝑥)𝑦𝜇(𝑥)𝑦𝑇𝜇(𝑥)𝜇𝑦(𝑥)𝑦𝑇𝜇(𝑥)𝜇(𝑥)𝑦𝑦𝑇𝜇(𝑥)(𝑥)𝜇𝑦𝑦𝜇(𝑥)𝑦,,𝜇,𝑇,𝜇=𝑇,𝑇,𝜇,𝜇=𝑇,𝑇,𝜇,𝜇=𝑇,𝜇,𝑇,𝜇=𝑇,𝜇 (7) and so 𝑇(𝜇(𝑥),𝜇(𝑦))∈Δ_𝑇. Assume that 𝑎=𝑇(𝜇(𝑥),𝜇(𝑦)). If 𝑎=0, then 𝑇𝜇(𝑥),𝜇𝑦=0≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧,𝑇𝜇(𝑥)𝑦),𝜇=0≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧). (8) Now, let 0<𝑎=𝑇(𝜇(𝑥),𝜇(𝑦))≤𝜇(𝑥)∧𝜇(𝑦)≤𝜇(𝑥)≤𝑏. Hence 𝑥,𝑦∈𝜇⁻¹[𝑎,𝑏], which implies that 𝑥+𝑦⊆𝜇⁻¹[𝑎,𝑏] and 𝑥𝛾𝑦⊆𝜇⁻¹[𝑎,𝑏]. Therefore 𝑇(𝜇(𝑥),𝜇(𝑦))≤inf_{𝑧∈𝑥+𝑦}{𝜇(𝑧)} and 𝑇(𝜇(𝑥),𝜇(𝑦))≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇(𝑧)}.

Definition 6. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a fuzzy subset of 𝑅. Then(1)𝜇 is called a 𝑇-fuzzy left Γ-hyperideal of 𝑅 if 𝑇𝜇(𝑥),𝜇𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧𝜇𝑦),∀𝑥,𝑦∈𝑅,≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧),∀𝑥,𝑦∈𝑅,∀𝛾∈Γ. (9)(2)𝜇 is called a 𝑇-fuzzy right Γ-hyperideal of 𝑅 if 𝑇𝜇(𝑥),𝜇𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧𝜇(𝑥)),∀𝑥,𝑦∈𝑅,≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧),∀𝑥,𝑦∈𝑅,∀𝛾∈Γ. (10)(3)𝜇 is called a 𝑇-fuzzy Γ-hyperideal of 𝑅 if it is both a 𝑇-fuzzy left Γ-hyperideal and a 𝑇-fuzzy right Γ-hyperideal of 𝑅.

Theorem 7. Let 𝑅 be a Γ-semihyperring, 𝑇 be a 𝑡-norm, and 𝜇 be a fuzzy subset of 𝑆 with imaginable property and 𝑏 the maximum of Im 𝜇. Then, the following two statements are equivalent:(1)𝜇 is a 𝑇-fuzzy Γ-hyperideal of 𝑅,(2)𝜇⁻¹[𝑎,𝑏] is a Γ-hyperideal of 𝑅 whenever 𝑎∈Δ_𝑇 and 0<𝑎≤𝑏.

Proof. The proof is similar to the proof of Theorem 5.

Let 𝜇 be a fuzzy subset of 𝑅 and 𝑡∈[0,1]. The set 𝑈(𝜇;𝑡)={𝑥∈𝑅∣𝜇(𝑥)≥𝑡} is called a level subset of 𝜇. So, we obtain the following corollary.

Corollary 8. Let 𝑅 be a Γ-semihyperring and 𝜇 be a fuzzy subset of 𝑅. Then(1)𝜇 is a Min-fuzzy sub-Γ-semihyperring of 𝑅 if and only if every nonempty level subset is a sub-Γ-semihyperring of 𝑅;(2)𝜇 is a Min-fuzzy Γ-hyperideal of 𝑅 if and only if every nonempty level subset is a Γ-hyperideal of 𝑅.

Corollary 9. Let 𝐴 be a subset of 𝑅. Then(1)the characteristic function 𝜒_𝐴 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅 if and only if 𝐴 is a sub-Γ-semihyperring of 𝑅;(2)the characteristic function 𝜒_𝐴 is a 𝑇-fuzzy Γ-hyperideal of 𝑅 if and only if 𝐴 is a Γ-hyperideal of 𝑅.

Theorem 10. Let 𝑅 be a Γ-semihyperring and 𝐾 be a sub-Γ-semihyperring of 𝑅. Let 𝑇 be the 𝑡-norm defined by 𝑇(𝑎,𝑏)=max{0,𝑎+𝑏−1} and 𝜇 be a fuzzy subset of 𝑅 defined by 𝜇(𝑥)=𝑟𝑖𝑓𝑥∈𝐾𝑠𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒, (11) for all 𝑎,𝑏∈[0,1] and 𝑥∈𝑅, where 𝑟,𝑠∈[0,1] such that 𝑠<𝑟. Then, 𝜇 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅. In particular, if 𝑟=1 and 𝑠=0, then 𝜇 is imaginable.

Proof. The proof is similar to the proof of Theorem  2.6 in [29].

Definition 11. Let 𝑅₁ and 𝑅₂ be Γ₁ and Γ₂-semihyperrings, respectively. If there exists a map 𝜑∶𝑅₁→𝑅₂ and a bijection 𝑓∶Γ₁→Γ₂ such that 𝜑=𝜑(𝑧)(𝑥)𝑦,𝜑=𝜑(𝑧)(𝑥)𝑓𝛾𝜑𝑦,𝑥+𝑦∣𝑧∈𝑥+𝑦=𝜑+𝜑𝑥𝛾𝑦∣𝑧∈𝑥𝛾𝑦=𝜑 (12) for all 𝑥,𝑦∈𝑅₁ and 𝛾∈Γ, then we say (𝜑,𝑓) is a homomorphism from 𝑅₁ to 𝑅₂. Also, if 𝜑 is a bijection then (𝜑,𝑓) is called an isomorphism and 𝑅₁ and 𝑅₂ are isomorphic.

Proposition 12. Let 𝑅₁ and 𝑅₂ be Γ₁ and Γ₂-semihyperrings, respectively. Let (𝜑,𝑓) be a homomorphism from 𝑅₁ to 𝑅₂. If 𝜆 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅₂, then 𝜑⁻¹(𝜆) is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅₁ too.

Proof. Suppose that 𝑥,𝑦∈𝑅₁ and 𝛾∈Γ. Then, we have inf_{𝑧∈𝑥+𝑦}𝜑⁻¹(𝜆)(𝑧)=inf_{𝑧∈𝑥+𝑦}𝜆𝜑(𝑧)≥inf_{𝜑(𝑧)∈𝜑(𝑥+𝑦)}𝜆𝜑(𝑧)≥inf_{𝜑(𝑧)∈𝜑(𝑥)+𝜑(𝑦)}𝜆𝜑(𝑧)𝜆𝜑(𝑥)𝜑𝑦𝜑≥𝑇,𝜆=𝑇⁻¹(𝜆)(𝑥),𝜑⁻¹(𝜆)𝑦;inf_{𝑧∈𝑥𝛾𝑦}𝜑⁻¹(𝜆)(𝑧)=inf_{𝑧∈𝑥𝛾𝑦}𝜆𝜑(𝑧)≥inf_{𝜑(𝑧)∈𝜑(𝑥𝛾𝑦)}𝜆𝜑(𝑧)≥inf_{𝜑(𝑧)∈𝜑(𝑥)𝑓(𝛾)𝜑(𝑦)}𝜆𝜑(𝑧)𝜆𝜑(𝑥)𝜑𝑦𝜑≥𝑇,𝜆=𝑇⁻¹(𝜆)(𝑥),𝜑⁻¹(𝜆)𝑦. (13) Therefore, 𝜑⁻¹(𝜆) is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅₁.

Proposition 13. Let 𝑅₁ and 𝑅₂ be Γ₁ and Γ₂-semihyperrings, respectively. Let (𝜑,𝑓) be a homomorphism from 𝑅₁ to 𝑅₂. If 𝜆 is a 𝑇-fuzzy Γ-hyperideal of 𝑅₂, then 𝜑⁻¹(𝜆) is a 𝑇-fuzzy Γ-hyperideal of 𝑅₁ too.

Proof. The proof is similar to the proof of Proposition 12.

Let {𝑎_𝑖}_𝑖∈𝐼 and {𝑏_𝑗}_𝑗∈𝐽 be two sets of real numbers in [0,1]. Then, we say 𝑇 is infinitely distributive if 𝑇sup_𝑖∈𝐼𝑎_𝑖,sup_𝑗∈𝐽𝑏_𝑗=sup_{𝑖∈𝐼𝑗∈𝐽}𝑇𝑎_𝑖,𝑏_𝑗. (14) If 𝑇 is continuous, then 𝑇 is infinitely distributive [30].

Lemma 14. Let 𝑇 be a continuous 𝑡-norm and {𝜇_𝑖}_𝑖∈𝐼 be a family of 𝑇-fuzzy sub-Γ-semihyperring of 𝑅. Then, ⋂_𝑖∈𝐼𝜇_𝑖 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅.

Proof. For any 𝑥,𝑦∈𝑅 and 𝛾∈Γ, we have inf_{𝑧∈𝑥+𝑦𝑖∈𝐼}𝜇_𝑖(𝑧)=inf_{𝑧∈𝑥+𝑦}inf_𝑖∈𝐼𝜇_𝑖(𝑧)=inf_𝑖∈𝐼inf_{𝑧∈𝑥+𝑦}𝜇_𝑖(𝑧)≥inf_𝑖∈𝐼𝑇𝜇_𝑖(𝑥),𝜇_𝑖𝑦=𝑇inf_𝑖∈𝐼𝜇_𝑖(𝑥),inf_𝑖∈𝐼𝜇_𝑖𝑦=𝑇_𝑖∈𝐼𝜇_𝑖(𝑥),_𝑖∈𝐼𝜇_𝑖𝑦;inf_{𝑧∈𝑥𝛾𝑦𝑖∈𝐼}𝜇_𝑖(𝑧)=inf_{𝑧∈𝑥𝛾𝑦}inf_𝑖∈𝐼𝜇_𝑖(𝑧)=inf_𝑖∈𝐼inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝑖(𝑧)≥inf_𝑖∈𝐼𝑇𝜇_𝑖(𝑥),𝜇_𝑖𝑦=𝑇inf_𝑖∈𝐼𝜇_𝑖(𝑥),inf_𝑖∈𝐼𝜇_𝑖𝑦=𝑇_𝑖∈𝐼𝜇_𝑖(𝑥),_𝑖∈𝐼𝜇_𝑖𝑦. (15)

Lemma 15. Let 𝑅₁ and 𝑅₂ be Γ₁ and Γ₂-semihyperrings, respectively, and (𝜑,𝑓) be an onto homomorphism from 𝑅₁ to 𝑅₂. Then, for every 𝑡∈[0,1], we have 𝜑(𝜇)_𝑡=⋂_𝑡>𝜖>0𝜑(𝜇_𝑡−𝜖).

Proof. The proof is similar to the proof of Lemma  3.5 in [31].

Proposition 16. Let 𝑅₁ and 𝑅₂ be Γ₁ and Γ₂-semihyperrings, respectively, and let 𝜇 be a fuzzy subset of 𝑅₁. Let (𝜑,𝑓) be an onto homomorphism from 𝑅₁ to 𝑅₂. If 𝜇 is a Min-fuzzy sub-Γ-semihyperring of 𝑅₁, then 𝜑(𝜇) is a Min-fuzzy sub-Γ-semihyperring of 𝑅₂ too.

Proof. Suppose that 𝜇 is a Max-fuzzy sub-Γ-semihyperring of 𝑅₁. By Corollary 8, 𝜑(𝜇) is a Max-fuzzy sub-Γ-semihyperring of 𝑅₂ if every nonempty level subset 𝑈(𝜑((𝜇);𝑡)) is a sub-Γ-semihyperring of 𝑅₂. Thus, assume that 𝑈(𝜑((𝜇);𝑡)) is any nonempty level subset. If 𝑡=0, then 𝑈(𝜑((𝜇);𝑡))=𝑅₂, and if 0<𝑡≤1, then by Lemma 15, we have 𝑈(𝜑((𝜇);𝑡))=⋂_𝑡>𝜖>0𝜑(𝑈(𝜇;𝑡−𝜖)). By Corollary 8, 𝑈(𝜇;𝑡−𝜖) for each 𝑡>𝜖>0 is a sub-Γ-semihyperring of 𝑅₁. Hence, 𝜑(𝑈(𝜇;𝑡−𝜖)) is a sub-Γ-semihyperring of 𝑅₂. By Lemma 14, 𝜑(𝑈(𝜇;𝑡)) being an intersection of a family of sub-Γ-semihyperrings is also a sub-Γ-semihyperring of 𝑅₂ and the proof is completed.

Definition 17. Let 𝑅₁ and 𝑅₂ be two Γ-semihyperrings and let 𝜇 and 𝜆 be fuzzy subsets of 𝑅₁ and 𝑅₂, respectively. The product of 𝜇 and 𝜆 is defined to be the fuzzy subset 𝜇×𝜆 of 𝑅₁×𝑅₂ with (𝜇×𝜆)(𝑥,𝑦)=𝑇(𝜇(𝑥),𝜆(𝑥)), for all (𝑥,𝑦)∈𝑅₁×𝑅₂.

Proposition 18. Let 𝑅₁ and 𝑅₂ be two Γ-semihyperrings and let 𝜇 and 𝜆 be fuzzy subsets of 𝑅₁ and 𝑅₂, respectively. Then(1)if 𝜇 and 𝜆 are 𝑇-fuzzy sub-Γ-semihyperrings of 𝑅₁ and 𝑅₂, respectively, then 𝜇×𝜆 is a 𝑇-fuzzy sub-Γ-semihyperring of 𝑅₁×𝑅₂; (2)if 𝜇 and 𝜆 are 𝑇-fuzzy Γ-hyperideals of 𝑅₁ and 𝑅₂, respectively, then 𝜇×𝜆 is a 𝑇-fuzzy Γ-hyperideal of 𝑅₁×𝑅₂.

Proof. It is straightforward.

In [12], Dehkordi and Davvaz studied Noetherian and Artinian Γ-semihyperrings in crisp case. A collection 𝒜 of subsets of a Γ-semihyperring 𝑅 satisfies the ascending chain condition (or Acc) if there does not exist a properly ascending infinite chain 𝐴₁⊂𝐴₂⊂⋯ of subsets from 𝒜. Recall that a subset 𝐵∈𝒜 is a maximal element of 𝒜 if there does not exist a subset in 𝒜 that properly contains 𝐵. Similar to [18], in the following, we obtain some results related to fuzzy sets and Noetherian Γ-semihyperrings.

Proposition 19 (see [12]). Let 𝑅 be a Γ-semihyperring. Then, the following conditions are equivalent:(1)𝑅 satisfying the Acc condition on right (left) Γ-hyperideals, (2)every nonempty family of right (left) Γ-hyperideals has a maximal element, (3)every right (left) Γ-hyperideals is finitely generated.

Definition 20 (see [12]). A Γ-semihyperring 𝑅 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 𝐼 be a Γ-hyperideal of a Γ-semihyperring 𝑅 and 𝐼 be a Noetherian Γ-semihyperring. Then, 𝐼 is called a Noetherian Γ-hyperideal of 𝑅.

Example 21 (see [12]). Let 𝐴_𝑛=[𝑛,𝑛_𝑛+1] for every 𝑛∈ℤ, 𝑅=⋃_𝑛∈ℤ𝐴_𝑛, and Γ=ℤ. Then, 𝑅 is a Noetherian Γ-semihyperring with respect to the following hyperoperations: 𝑥⊕𝑦=𝐴_𝑛+𝑚,𝑥𝛾𝑦=𝐴_𝑛𝛾𝑚, (16) where 𝑥∈𝐴_𝑛 and 𝑦∈𝐴_𝑚.

Theorem 22. Let {𝐴_𝑘∣𝑘∈ℕ} be a family of Γ-hyperideals of a Γ-semihyperring 𝑅, where 𝐴₁⊃𝐴₂⊃𝐴₃⋯. Let 𝜇 be a fuzzy subset of 𝑅 defined by 𝜇(𝑥)=𝑘𝑘+1𝑖𝑓𝑥∈𝐴_𝑘⧵𝐴_𝑘+1,𝑘=0,1,2,…1𝑖𝑓𝑥∈^∞_𝑘=0𝐴_𝑘, (17) for all 𝑥∈𝑅, where 𝐴₀ stands for 𝑅. Let 𝑇 be a t-norm with Im 𝜇⊆Δ_𝑇. Then, 𝜇 is a 𝑇-fuzzy Γ-hyperideal of 𝑅.

Proof. Let 𝑥,𝑦∈𝑅. Suppose that 𝑥∈𝐴_𝑘⧵𝐴_𝑘+1 and 𝑦∈𝐴_𝑟⧵𝐴_𝑟+1 for 𝑘=0,1,2,… and 𝑟=0,1,2,…. Without loss of generality we may assume that 𝑘≤𝑟. Then, obviously 𝑦∈𝐴_𝑘. Since 𝐴_𝑘 is a Γ-hyperideal of 𝑅, it follows that 𝑥+𝑦⊆𝐴_𝑘 and 𝑥𝛼𝑦⊆𝐴_𝑘 which imply that inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥𝑘/(𝑘+1)=𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)≥𝑘/(𝑘+1)=𝜇(𝑦) for all 𝛼∈Γ. If 𝑥∈⋂^∞_𝑘=0𝐴_𝑘 and 𝑦∈⋂^∞_𝑘=0𝐴_𝑘, then 𝑥+𝑦⊆⋂^∞_𝑘=0𝐴_𝑘. Hence, inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)=1=𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)=1≥𝜇(𝑦). If 𝑥∉⋂^∞_𝑘=0𝐴_𝑘 and 𝑦∈⋂^∞_𝑘=0𝐴_𝑘, then there exists 𝑛∈𝑁 such that 𝑥∈𝐴_𝑛⧵𝐴_𝑛+1. It follows that 𝑥+𝑦⊆𝐴_𝑛 which implies that inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥𝑛/(𝑛+1)=𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)≥𝑛/(𝑛+1)=𝜇(𝑦) for all 𝛼∈Γ. Finally, assume that 𝑥∈⋂^∞_𝑘=0𝐴_𝑘 and 𝑦∉⋂^∞_𝑘=0𝐴_𝑘, then 𝑦∈𝐴_𝑛⧵𝐴_𝑛+1 for some 𝑛∈ℕ. Therefore, 𝑥+𝑦⊆𝐴_𝑛, and thus inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥𝑛/(𝑛+1)=𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)≥𝑛/(𝑛+1)=𝜇(𝑦), for all 𝛼∈Γ. Hence, 𝜇 is a 𝑇-fuzzy Γ-hyperideal of 𝑅.

Theorem 23. Let 𝑅 be a Γ-semiring satisfying descending chain condition, let 𝜇 be a fuzzy subset of 𝑅, and let 𝑇 be a 𝑡-norm with Im 𝜇⊆Δ_𝑇. Let 𝜇 be a 𝑇-fuzzy Γ-hyperideal of 𝑅. If a sequence of elements of Im 𝜇 is strictly increasing, then 𝜇 has finite number of values.

Proof. Let {𝑡_𝑘} be a strictly increasing sequence of elements of Im𝜇. Then 0≤𝑡₁≤𝑡₂≤⋯≤1. Then, 𝑈(𝜇;𝑡_𝑟) is an ideal of 𝑀 for all 𝑟=2,3,… Let 𝑥∈𝑈(𝜇;𝑡_𝑟). Then 𝜇(𝑥)≥𝑡_𝑟≥𝑡_𝑟−1, and so 𝑥∈𝑈(𝜇;𝑡_𝑟−1). Hence 𝑈(𝜇;𝑡_𝑟)⊆𝑈(𝜇;𝑡_𝑟−1). Since 𝑡_𝑟−1∈Im𝜇, there exists 𝑥_𝑟−1∈𝑀 such that 𝜇(𝑥_𝑟−1)=𝑡_𝑟−1. It follows that 𝑥_𝑟−1∈𝑈(𝜇;𝑡_𝑟−1) but 𝑥_𝑟−1∉𝑈(𝜇;𝑡_𝑟). Thus, 𝑈(𝜇;𝑡_𝑟)⊂𝑈(𝜇;𝑡_𝑟−1) and so we obtain a strictly descending sequence 𝑈(𝜇;𝑡₁)⊃𝑈(𝜇;𝑡₂)⊃𝑈(𝜇;𝑡₃)⊃⋯ of Γ-hyperideals of 𝑅 which is not terminating. This contradicts the assumption that 𝑀 satisfies the descending chain condition. Consequently, 𝜇 has finite number of values.

Theorem 24. Let 𝑅 be a Γ-semiring, 𝜇 be a fuzzy subset of 𝑅, and 𝑇 be a 𝑡-norm with Im 𝜇⊆Δ_𝑇. Then, the following conditions are equivalent: (1)𝑅 is a Noetherian Γ-semihyperring,(2)the set of values of any Γ-hyperideal of 𝑅 is a well-ordered subset of [0,1].

Proof. (1)⇒(2): Let 𝜇 be a 𝑇-fuzzy Γ-hyperideal of 𝑅. Suppose that the set of values of 𝜇 is not a well-ordered subset of [0,1]. Then, there exists a strictly decreasing sequence {𝑡_𝑘} such that 𝜇(𝑥_𝑘)=𝑡_𝑘. It follows that 𝑈(𝜇;𝑡₁)⊂𝑈(𝜇;𝑡₂)⊂⋯ is a strictly ascending chain of Γ-hyperideals of 𝑀, where 𝑈(𝜇;𝑡_𝑟)={𝑥∈𝑀∣𝜇(𝑥)≥𝑡_𝑟}, for every 𝑟=1,2,…. This contradicts the assumption that 𝑅 is a Noetherian Γ-semihyperring.
(2)⇒(1): Suppose that the condition (2) is satisfied and 𝑅 is not a Noetherian Γ-hyperring. There exists a strictly ascending chain 𝐴₁⊂𝐴₂⊂𝐴₃⊂⋯ (18) of Γ-hyperideals of 𝑅. Note that 𝐴=⋂^∞_𝑘=0𝐴_𝑘 is a Γ-hyperideal of 𝑅. Define a fuzzy subset in 𝑅 by 𝜇(𝑥)=𝛼if1𝑥∉𝐴,𝑟where𝑟=min𝑘∈ℕ∣𝑥∈𝐴_𝑘. (19) We claim that 𝜇 is a 𝑇-fuzzy Γ-hyperideal of 𝑅. Let 𝑥,𝑦∈𝑅. If 𝑥∈𝐴_𝑘⧵𝐴_𝑘−1 and 𝑦∈𝐴_𝑘⧵𝐴_𝑘−1, then 𝑥−𝑦⊂𝐴_𝑘 and 𝑥𝛼𝑦⊂𝐴_𝑘. It follows that inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)=1/𝑘=𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)≥1/𝑘≥𝜇(𝑦), for all 𝛽∈Γ. Suppose that 𝑥∈𝐴_𝑘 and 𝑦∈𝐴_𝑘⧵𝐴_𝑟 for all 𝑟<𝑘. Since 𝐴_𝑘 is a Γ-hyperideal of 𝑅, it follows that 𝑥+𝑦⊂𝐴_𝑘. Hence, inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥1/𝑘≥1/(𝑘+1)≥𝜇(𝑦) and so inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥min{𝜇(𝑥),𝜇(𝑦)}≥𝑇(𝜇(𝑥),𝜇(𝑦)). Similarly, for the case 𝑥∈𝐴_𝑘⧵𝐴_𝑟 and 𝑦∈𝐴_𝑘, we have inf_{𝑧∈𝑥+𝑦}𝜇(𝑧)≥min{𝜇(𝑥),𝜇(𝑦)}≥𝑇(𝜇(𝑥),𝜇(𝑦)) and inf_{𝑧∈𝑥𝛼𝑦}𝜇(𝑧)≥𝜇(𝑦), for all 𝛽∈Γ. Thus, 𝜇 is a 𝑇-fuzzy Γ-hyperideal of 𝑅. Since the chain is not terminating, 𝜇 has a strictly descending sequence of values. This contradicts the assumption that the value set of any ideal is well-ordered. Therefore, 𝑅 is a Noetherian Γ-semihyperring.

For a family {𝜇_𝛼∣𝛼∈Λ} of fuzzy subsets in 𝑅, we define the join ∨_𝛼∈Λ𝜇_𝛼 and the meet ∨_𝛼∈Λ𝜇_𝛼 as follows: _𝛼∈Λ𝜇_𝛼(𝑥)=sup𝜇_𝛼(𝑥)∣𝛼∈Λ_𝛼∈Λ,𝜇_𝛼(𝑥)=inf𝜇_𝛼(𝑥)∣𝛼∈Λ, (20) for all 𝑥∈𝑅, where Λ is any index set.

Theorem 25. The family of 𝑇-fuzzy Γ-hyperideals in 𝑅 is a completely distributive lattice with respect to meet “∧” and join “∨”.

Proof. Since [0,1] is a completely distributive lattice with respect to the usual ordering in [0,1], it is sufficient to show that ∨_𝛼∈Λ𝜇_𝛼 and ∧_𝛼∈Λ𝜇_𝛼 are 𝑇-fuzzy Γ-hyperideals of 𝑅 for family {𝜇_𝛼∣𝛼∈Λ} of 𝑇-fuzzy Γ-hyperideals of 𝑅. For any 𝑥,𝑦∈𝑅, we have inf_{𝑧∈𝑥+𝑦𝛼∈Λ}𝜇_𝛼(𝑧)≥sup𝑇𝜇_𝛼(𝑥),𝜇_𝛼𝑦𝜇∣𝛼∈Λ≥𝑇sup_𝛼(𝑥),𝜇∣𝛼∈Λsup_𝛼𝑦∣𝛼∈Λ=𝑇_𝛼∈Λ𝜇_𝛼(𝑥),_𝛼∈Λ𝜇_𝛼𝑦,inf_{𝑧∈𝑥+𝑦𝛼∈Λ}𝜇_𝛼(𝑧)≥inf𝑇𝜇_𝛼(𝑥),𝜇_𝛼𝑦∣𝛼∈Λ=𝑇_𝛼∈Λ𝜇_𝛼(𝑥),_𝛼∈Λ𝜇_𝛼𝑦. (21)
Now, let 𝑥,𝑦∈𝑀 and 𝛽∈Γ. Then inf_{𝑧∈𝑥𝛽𝑦𝛼∈Λ}𝜇_𝛼(𝑧)≥sup𝜇_𝛼𝑦=∣𝛼∈Λ_𝛼∈Λ𝜇_𝛼𝑦,inf_{𝑧∈𝑥𝛽𝑦𝛼∈Λ}𝜇_𝛼(𝑧)≥inf𝜇_𝛼𝑦=∣𝛼∈Λ_𝛼∈Λ𝜇_𝛼𝑦. (22)(23) Hence, ∨_𝛼∈Λ𝜇_𝛼 and ∧_𝛼∈Λ𝜇_𝛼 are 𝑇-fuzzy Γ-hyperideals of 𝑅. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.