Structure of Intuitionistic Fuzzy Sets in Γ-Semihyperrings

Abstract

As we know, intuitionistic fuzzy sets are extensions of the standard fuzzy sets. Now, in this paper, the basic definitions and properties of intuitionistic fuzzy Γ-hyperideals of a Γ-semihyperring are introduced. A few examples are presented. In particular, some characterizations of Artinian and Noetherian Γ-semihyperring are given.

1. Introduction

The theory of fuzzy sets proposed by Zadeh [1] has achieved a great success in various fields. In 1971, Rosenfeld [2] introduced fuzzy sets in the context of group theory and formulated the concept of a fuzzy subgroup of a group. Since then, many researchers are engaged in extending the concepts of abstract algebra to the framework of the fuzzy setting.

The concept of a fuzzy ideal of a ring was introduced by Liu [3]. The concept of intuitionistic fuzzy set was introduced and studied by Atanassov [4–6] as a generalization of the notion of fuzzy set. In [7], Biswas studied the notion of an intuitionistic fuzzy subgroup of a group. In [8], Kim and Jun introduced the concept of intuitionistic fuzzy ideals of semirings. Also, in [9], Gunduz and Davvaz studied the universal coefficient theorem in the category of intuitionistic fuzzy modules. Also see [10, 11].

In 1964, Nobusawa introduced Γ-rings as a generalization of ternary rings. Barnes [12] weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes [12], Luh [13], and Kyuno [14] studied the structure of Γ-rings and obtained various generalization analogous to corresponding parts in ring theory. The concept of Γ-semigroups was introduced by Sen and Saha [15, 16] as a generalization of semigroups and ternary semigroups. Then the notion of Γ-semirings introduced by Rao [17].

Algebraic hyperstructures represent a natural extension of classical algebraic structures, and they were introduced by the French mathematician Marty [18]. 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 hyperstructure, the composition of two elements is a set. Since then, hundreds of papers and several books have been written on this topic, see [19–22].

In [23–25], Davvaz et al. introduced the notion of a Γ-semihypergroup as a generalization of a semihypergroup. Many classical notions of semigroups and semihypergroups have been extended to Γ-semihypergroups and a lot of results on Γ-semihypergroups are obtained. In [26–29], Davvaz et al. studied the notion of a Γ-semihyperring as a generalization of semiring, a generalization of a semihyperring, and a generalization of a Γ-semiring.

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, see [20]. In [30], Davvaz introduced the notion of fuzzy subhypergroups as a generalization of fuzzy subgroups, and this topic was continued by himself and others. In [31], Leoreanu-Fotea and Davvaz studied fuzzy hyperrings. Recently, Davvaz et al. [32–35] considered the intuitionistic fuzzification of the concept of algebraic hyperstructurs and investigated some properties of such hyperstructures.

Now, in this work we introduce the notion of an Atanassov’s intuitionistic fuzzy hyperideals of a semihyperrings and investigate some basic properties about it.

2. Basic Definitions

Let 𝑆 be a nonempty set, and let ℘^•(𝑆) be the set of all non-empty subsets of 𝑆. A hyperoperation on 𝑆 is a map ∘∶𝑆×𝑆→℘^•(𝑆), and the couple (𝑆,∘) is called a hypergrupoid. If 𝐴 and 𝐵 are non-empty subsets of 𝑆, then we denote 𝐴∘𝐵=∪_{𝑎∈𝐴,𝑏∈𝐵}{𝑥}{𝑥}.𝑎∘𝑏,𝑥∘𝐴=∘𝐴,𝐴∘𝑥=𝐴∘ (1)

A hypergrupoid (𝑆,∘) is called a semihypergroup if for all 𝑥,𝑦,and𝑧 of 𝑆 we have (𝑥∘𝑦)∘𝑧=𝑥∘(𝑦∘𝑧). That is, _𝑢∈∪_𝑥∘𝑦𝑢∘𝑧=∪_{𝑣∈𝑦∘𝑧}𝑥∘𝑣. (2) A semihypergroup (𝑆,∘) is called a hypergroup if for all 𝑥∈𝑆, 𝑥∘𝑆=𝑆∘𝑥=𝑆.

A semihyperring is an algebraic structure (𝑆,+,⋅) which satisfies the following properties:(1)(𝑆,+) is a commutative semihypergroup; that is,(i)(𝑥+𝑦)+𝑧=𝑥+(𝑦+𝑧), (ii)𝑥+𝑦=𝑦+𝑥,  for all 𝑥,𝑦,𝑧∈𝑆.(2)(𝑆,⋅) is a semihypergroup; that is, (𝑥⋅𝑦)⋅𝑧=𝑥⋅(𝑦⋅𝑧), for all 𝑥,𝑦,𝑧∈𝑆.(3) The multiplication is distributive with respect to hyperoperation +; that is, 𝑥⋅𝑦+𝑧=𝑥⋅𝑦+𝑥⋅𝑧,𝑥+𝑦⋅𝑧=𝑥⋅𝑧+𝑦⋅𝑧, (3) for all 𝑥,𝑦,𝑧∈𝑆.(4) The element 0∈𝑆 is an absorbing element; that is, 0⋅𝑥=𝑥⋅0=0, for all 𝑥∈𝑆. A semihyperring (𝑆,+,⋅) is called commutative if 𝑥⋅𝑦=𝑦⋅𝑥 for all 𝑥,𝑦∈𝑆. Vougiouklis in [22] studied the notion of semihyperrings in a general form. That is, both sum and multiplication are hyperoperations.

A semihyperring 𝑆 has identity element if there exists 1_𝑆∈𝑆, such that 1_𝑆⋅𝑥=𝑥⋅1_𝑆=𝑥, for all 𝑥∈𝑆. An element 𝑥∈𝑆 is called unit if there exists 𝑦∈𝑆, such that 𝑦⋅𝑥=𝑥⋅𝑦=1_𝑆. A non-empty subset 𝐴 of a semihyperring (𝑆,+,⋅) is called subsemihyperring if 𝑥+𝑦⊆𝐴 and 𝑥⋅𝑦⊆𝐴, for all 𝑥,𝑦∈𝐴. A left hyperideal of a semihyperring 𝑆 is a non-empty subset 𝐼 of 𝑆 satisfying the following:(i)𝑥+𝑦⊆𝐼, for all 𝑥,𝑦∈𝐼,(ii)𝑥⋅𝑎⊆𝐼, for all 𝑎∈𝐼 and 𝑥∈𝑆.

Let 𝑆 and Γ be two non-empty sets. Then, 𝑆 is called a Γ-semihypergroup; if for every hyperoperation 𝛾∈Γ, 𝛼,𝛽∈Γ, and 𝑥,𝑦,𝑧∈𝑆, we have 𝑥𝛽𝑦𝛾𝑧=𝑥𝛽𝑦𝛾𝑧. (4)

For example, 𝑆=[0,1] and Γ⊆𝑆. For all 𝛼∈Γ and for all 𝑥,𝑦∈𝑆, we define 𝑥𝛼𝑦=[0,max{𝑥,𝛼,𝑦}]. Then, 𝑆 is a Γ-semihypergroup.

The concept of Γ-semihyperring was introduced and studied by Dehkordi and Davvaz [26–28]. We recall the following definition from [26].

Definition 1. Let 𝑆 be a commutative semihypergroup, and Γ be a commutative group. Then, 𝑆 is called a Γ-semihyperring if there exists a map 𝑆×Γ×𝑆→℘^∗(𝑆) (image to denoted by 𝑥𝛾𝑦) satisfying the following conditions: (i)𝑥𝛼(𝑦+𝑧)=𝑥𝛼𝑦+𝑥𝛼𝑧, (ii)(𝑥+𝑦)𝛼𝑧=𝑥𝛼𝑧+𝑦𝛼𝑧, (iii)𝑥(𝛼+𝛽)𝑧=𝑥𝛼𝑧+𝑥𝛽𝑧, (iv)𝑥𝛼(𝑦𝛽𝑧)=(𝑥𝛼𝑦)𝛽𝑧,  for all 𝑥,𝑦,𝑧∈𝑆 and 𝛼,𝛽∈Γ.

In the above definition, if 𝑆 is a semigroup, then 𝑆 is called a multiplicative Γ-semihyperring. A Γ-semihyperring 𝑆 is called commutative if 𝑥𝛼𝑦=𝑦𝛼𝑥, for all 𝑥,𝑦∈𝑆 and 𝛼∈Γ. We say that a Γ-semihyperring 𝑆 is with zero, if there exists 0∈𝑆, such that 𝑥∈𝑥+0 and 0∈0𝛼𝑥, for all 𝑥∈𝑆 and 𝛼∈Γ. Let 𝐴 and 𝐵 be two non-empty subsets of a Γ-semihyperring and 𝑥∈𝑆. We define {},{}.𝐴+𝐵=𝑥∣𝑥∈𝑎+𝑏,𝑎∈𝐴,𝑏∈𝐵𝐴Γ𝐵=𝑥∣𝑥∈𝑎𝛼𝑏,𝑎∈𝐴,𝑏∈𝐵,𝛼∈Γ (5)

A non-empty subset of 𝑆₁ of Γ-semihyperring 𝑆 is called a 𝑠𝑢𝑏-Γ-semihyperring if it is closed with respect to the multiplication and addition. In other words, a non-empty subset 𝑆₁ of Γ-semihyperring 𝑆 is a sub Γ-semihyperring if 𝑆₁+𝑆₁⊆𝑆₁,𝑆₁Γ𝑆₁⊆𝑆₁. (6)

Example 2. Let (𝑆,+,⋅) be a semiring, and let Γ be a subsemiring of 𝑆. We define 𝑥𝛾𝑦=𝑥,𝛾,𝑦, (7) the ideal generated by 𝑥,𝛾,and𝑦, for all 𝑥,𝑦∈𝑆 and 𝛾∈Γ. Then, it is not difficult to see that 𝑆 is a multiplicative Γ-semihyperring.

Example 3. Let 𝑆=ℚ⁺, Γ={𝛾_𝑖∣𝑖∈ℕ} and 𝐴_𝑖=𝑖ℤ⁺. We define 𝑥𝛾_𝑖𝑦→𝑥𝐴_𝑖𝑦 for every 𝛾∈Γ and 𝑥,𝑦∈𝑆. Then, 𝑆 is a Γ-semihyperring under ordinary addition and multiplication.

Example 4. Let (𝑆,+,⋅) be a semihyperring, and let 𝑀_𝑚,𝑛(𝑆) the set of all 𝑚×𝑛 matrices with entries in 𝑆. We define ∘∶𝑀_𝑚,𝑛(𝑆)×𝑀_𝑛,𝑚(𝑆)×𝑀_𝑚,𝑛(𝑆)→𝑃^∗(𝑀_𝑚,𝑛(𝑆)) by 𝐴∘𝐵∘𝐶=𝑍∈𝑀_𝑚,𝑛(𝑆)∣𝑍∈𝐴𝐵𝐶, (8) for all 𝐴,𝐶∈𝑀_𝑚,𝑛(𝑆) and 𝐵∈𝑀_𝑛,𝑚(𝑆).
Then, 𝑀_𝑚,𝑛(𝑆) is 𝑀_𝑛,𝑚(𝑆)-semihyperring [29]. Now, suppose that 𝑋=𝑎_{𝑖𝑗𝑚,𝑛}∣𝑎₁₁≠0,𝑎₂₁≠0,𝑎_𝑖𝑗=0otherwise. (9) Now, it is easy to see that 𝑋 is a sub Γ-semihyperring of 𝑆.
A non-empty subset 𝐼 of a Γ-semihyperring 𝑆 is a left (right) Γ-hyperideal of 𝑆 if for any 𝐼₁,𝐼₂⊆𝐼 implies 𝐼₁+𝐼₂⊆𝐼 and 𝐼Γ𝑆⊆𝐼(𝑆Γ𝐼⊆𝐼) and is a Γ-hyperideal of 𝑆 if it is both left and right Γ-hyperideal.

Example 5. Consider the following: 𝑆=𝑎𝑏𝑐𝑑∣𝑎,𝑏,𝑐,𝑑∈ℤ⁺∪{0},𝐴_𝑖=𝑖𝑒00𝑖𝑓∣𝑒,𝑓∈2ℤ⁺𝛾,∀𝑖∈ℕ,Γ=_𝑖∣𝑖∈ℕ. (10) Then, 𝑆 is a Γ-semihyperring under the matrix addition and the hyperoperation as follows: 𝑥𝛾_𝑖𝑦=𝑥𝐴_𝑖𝑦, (11) for all 𝑥,𝑦∈𝑆 and 𝛾_𝑖∈Γ. Let 𝐼₁=𝑎𝑏00∣𝑎,𝑏∈ℤ⁺∪{0},𝐼₂=𝑎0𝑐0∣𝑎,𝑐∈ℤ⁺∪{0},𝐼₃=𝑎𝑏𝑐𝑑∣𝑎,𝑏,𝑐,𝑑∈2ℤ⁺∪{0}. (12) Then, 𝐼₁ is a right Γ-hyperideal of 𝑆, but not left. 𝐼₂ is a left Γ-hyperideal of 𝑆, but not right. 𝐼₃ is both right and left Γ-hyperideal of 𝑆.
A fuzzy subset 𝜇 of a non-empty set 𝑆 is a function from 𝑆 to [0,1]. For all 𝑥∈𝑆𝜇^𝑐, the complement of 𝜇 is the fuzzy subset defined by 𝜇^𝑐(𝑥)=1−𝜇(𝑥). The intersection and the union of two fuzzy subsets 𝜇 and 𝜆 of 𝑆, denoted by 𝜇∩𝜆 and 𝜇∪𝜆, are defined by (𝑥)𝜇(𝑥)(𝑥,𝑥)𝜇(𝑥)(𝑥.𝜇∩𝜆=min,𝜆)𝜇∪𝜆(=max,𝜆) (13)

Definition 6. Let  𝑆 be a Γ-semihyperring, and let 𝜇 be a fuzzy subset of 𝑆. Then,(1)𝜇 is called a fuzzy left Γ-hyperideal of 𝑆 if min𝜇(𝑥),𝜇𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧,𝜇𝑦)≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧)∀𝑥,𝑦∈𝑆,∀𝛾∈Γ. (14)(2)𝜇 is called a fuzzy right Γ-hyperideal of 𝑆 if min𝜇(𝑥),𝜇𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇(𝑧,𝜇(𝑥))≤inf_{𝑧∈𝑥𝛾𝑦}𝜇(𝑧)∀𝑥,𝑦∈𝑆,∀𝛾∈Γ. (15)(3)𝜇 is called a fuzzy Γ-hyperideal of 𝑆 if 𝜇 is both a fuzzy left Γ-hyperideal and fuzzy right Γ-hyperideal of 𝑆.

Example 7. Let 𝑆₁=ℤ₆, Γ={𝛾₂,𝛾₃}, 𝑆₂={0,3}, and 𝑆₃={0} be non-empty subsets of 𝑆₁. We define 𝑥𝛾_𝑖𝑦=𝑥𝑆_𝑖𝑦, for every 𝛾_𝑖∈Γ and 𝑥,𝑦∈𝑆₁. Then, 𝑆₁ is a Γ-semihypering. Now, we define the fuzzy subset 𝜇 of 𝑆₁ as follows: 𝜇(𝑥)=0if𝑥=1,2,4,15,3if𝑥=13,if𝑥=0. (16) It is easy to see that 𝜇 is a fuzzy Γ-hyperideal of 𝑆₁.

3. Atanassov’s Intuitionistic Fuzzy Γ-Hyperideals

The concept of intuitionistic fuzzy set was introduced and studied by Atanassov [4–6]. Intuitionistic fuzzy sets are extensions of the standard fuzzy sets. An intuitionistic fuzzy set 𝐴 in a non-empty set 𝑆 has the form 𝐴={(𝑥,𝜇_𝐴(𝑥),𝜆_𝐴(𝑥))∣𝑥∈𝑆}. Here, 𝜇_𝐴∶𝑆→[0,1] is the degree of membership of the element 𝑥∈𝑆 to the set 𝐴, and 𝜆_𝐴∶𝑆→[0,1] is the degree of nonmembership of the element 𝑥∈𝑆 to the set 𝐴. We have also 0≤𝜇_𝐴(𝑥)+𝜆_𝐴(𝑥)≤1, for all 𝑥∈𝑆.

Example 8 (see [35]). Consider the universe 𝑋={10,100,500,1000,1200}. An intuitionistic fuzzy set “Large” of 𝑋 denoted by 𝐴 and may be defined by ⟩,⟨⟩,⟨⟩,⟨⟩,⟨.𝐴={⟨10,0.01,0.9100,0.1,0.88500,0.4,0.051000,0.8,0.11200,1,0⟩} (17)
One may define an intuitionistic fuzzy set “Very Large” denoted by 𝐵, as follows: 𝜇_𝐵(𝑥)=𝜇_𝐴(𝑥)²,𝜆_𝐵(𝑥)=1−1−𝜆_𝐴(𝑥)², (18) for all 𝑥∈𝑋. Then, ⟩,⟨⟩,⟨⟩,⟨⟩,⟨.𝐵={⟨10,0.0001,0.99100,0.01,0.9856500,0.16,0.751000,0.64,0.191200,1,0⟩} (19) For the sake of simplicity, we shall use the symbol 𝐴=(𝜇_𝐴,𝜆_𝐴) instead of 𝐴={(𝑥,𝜇_𝐴(𝑥),𝜆_𝐴(𝑥))∣𝑥∈𝑆}. Let 𝐴=(𝜇_𝐴,𝜆_𝐴) and 𝐵=(𝜇_𝐵,𝜆_𝐵) be two intuitionistic fuzzy sets of 𝑆. Then, the following expressions are defined in [4] as follows:(1)𝐴⊆𝐵 if and only if 𝜇_𝐴(𝑥)≤𝜇_𝐵(𝑥) and 𝜆_𝐴(𝑥)≥𝜆_𝐵(𝑥), for all 𝑥∈𝑆,(2)𝐴^𝐶={(𝑥,𝜆_𝐴(𝑥),𝜇_𝐴(𝑥))∣𝑥∈𝑆},(3)𝐴∩𝐵={(𝑥,min{𝜇_𝐴(𝑥),𝜇_𝐵(𝑥)},max{𝜆_𝐴(𝑥),𝜆_𝐵(𝑥)})∣𝑥∈𝑆},(4)𝐴∪𝐵={(𝑥,max{𝜇_𝐴(𝑥),𝜇_𝐵(𝑥)},min{𝜆_𝐴(𝑥),𝜆_𝐵(𝑥)})∣𝑥∈𝑆},(5)□𝐴={(𝑥,𝜇_𝐴(𝑥),𝜇^𝑐_𝐴(𝑥))∣𝑥∈𝑆},(6)⋄𝐴={(𝑥,𝜆^𝑐_𝐴(𝑥),𝜆_𝐴(𝑥))∣𝑥∈𝑆}.
Now, we introduce the notion of intuitionistic fuzzy Γ-hyperideals of Γ-semihyperrings.

Definition 9. Let 𝑆 be a Γ-semihyperring. (1) An intuitionistic fuzzy set 𝐴=(𝜇_𝐴,𝜆_𝐴) in 𝑆 is called a left intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring 𝑆 if (i)min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}≤inf_{𝑧∈𝑥+𝑦}{𝜇_𝐴(𝑧)}  and max{𝜆_𝐴(𝑥),𝜆_𝐴(𝑦)}≥sup_{𝑧∈𝑥+𝑦}{𝜆_𝐴(𝑧)},(ii)𝜇_𝐴(𝑦)≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐴(𝑧)} and sup_{𝑧∈𝑥𝛾𝑦}{𝜆_𝐴(𝑧)}≤𝜆_𝐴(𝑦), for all 𝑥,𝑦∈𝑆,𝛾∈Γ.(2) An intuitionistic fuzzy set 𝐴=(𝜇_𝐴,𝜆_𝐴) in 𝑆 is called a right intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring 𝑆 if (i)min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}≤inf_{𝑧∈𝑥+𝑦}{𝜇_𝐴(𝑧)}  and max{𝜆_𝐴(𝑥),𝜆_𝐴(𝑦)}≥sup_{𝑧∈𝑥+𝑦}{𝜆_𝐴(𝑧)},(ii)𝜇_𝐴(𝑥)≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐴(𝑧)} and sup_{𝑧∈𝑥𝛾𝑦}{𝜆_𝐴(𝑧)}≤𝜆_𝐴(𝑥), for all 𝑥,𝑦∈𝑆,𝛾∈Γ.(3) An intuitionistic fuzzy set 𝐴=(𝜇_𝐴,𝜆_𝐴) in 𝑆 is called an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring 𝑆 if it is both left intuitionistic fuzzy Γ-hyperideal and right intuitionistic fuzzy Γ-hyperideal of 𝑆.

Example 10. Let 𝑆₁ be a Γ-semihyperring in Example 7. We define the fuzzy subsets 𝜇_𝐴 and 𝜆_𝐴 of 𝑆₁ as follows: 𝜇(𝑥)=0if𝑥=1,2,4,15,4if𝑥=23,5if𝑥=𝜆(𝑥)=10,2if𝑥=1,2,4,15,3if𝑥=33,10if𝑥=0. (20) Then, 𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆₁.

Example 11. Let 𝑆 be a Γ-semihyperring defined in Example 5. Suppose that 𝜇_𝐴𝑖(𝑥)=58if𝑎,𝑏,𝑐,𝑑∈2ℤ⁺∪{0},25otherwise,𝜆_𝐴𝑖(𝑥)=19if𝑎,𝑏,𝑐,𝑑∈2ℤ⁺∪{0},47otherwise, (21) for all 𝑥∈𝑆. Then, 𝐴_𝑖=(𝜇_𝐴𝑖,𝜆_𝐴𝑖) is an intuitionistic fuzzy Γ-hyperideal of 𝑆.

Theorem 12. Let 𝐴=(𝜇_𝐴,𝜆_𝐴) be an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring 𝑆 and 0≤𝑡≤1. We define an intuitionistic fuzzy set 𝐵=(𝜇_𝐵,𝜆_𝐵) in 𝑆 by 𝜇_𝐵(𝑥)=𝑡𝜇_𝐴(𝑥) and 𝜆_𝐵(𝑥)=(1−𝑡)𝜆_𝐵(𝑥), for all 𝑥∈𝑆. Then, 𝐵=(𝜇_𝐵,𝜆_𝐵) is an intuitionistic fuzzy Γ-hyperideal of 𝑆.

Proof. Note that 0≤𝜇_𝐵(𝑥)+𝜆_𝐵(𝑥)=𝑡𝜇_𝐴(𝑥)+(1−𝑡)𝜆_𝐴(𝑥)≤𝑡+1−𝑡=1.

Theorem 13. Let 𝐴=(𝜇_𝐴,𝜆_𝐴) be an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring 𝑆. We define an intuitionistic fuzzy set 𝐵=(𝜇_𝐵,𝜆_𝐵) in 𝑆 by 𝜇_𝐵(𝑥)=(𝜇_𝐴(𝑥))² and 𝜆_𝐵(𝑥)=1−(1−𝜆_𝐵(𝑥))², for all 𝑥∈𝑆. Then, 𝐵=(𝜇_𝐵,𝜆_𝐵) is an intuitionistic fuzzy Γ-hyperideals of 𝑆.

Proof. Since 𝐴=(𝜇_𝐴,𝜆_𝐴) is an intuitionistic fuzzy Γ-hyperideal, we have inf_{𝑧∈𝑥+𝑦}{𝜇_𝐴(𝑧)}≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)} and so inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧)²≥min𝜇_𝐴(𝑥),𝜇_𝐴𝑦². (22) Hence, inf_{𝑧∈𝑥+𝑦}{(𝜇_𝐴(𝑧))²}≥min{(𝜇_𝐴(𝑥))²,(𝜇_𝐴(𝑦))²}; that is, inf_{𝑧∈𝑥+𝑦}𝜇_𝐵(𝑧)≥min𝜇_𝐵(𝑥),𝜇_𝐵𝑦. (23) Since inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐴(𝑧)}≥𝜇_𝐴(𝑦), so (inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐴(𝑧)})²≥(𝜇_𝐴(𝑦))² which implies that inf_{𝑧∈𝑥𝛾𝑦}{(𝜇_𝐴(𝑧))²}≥(𝜇_𝐴(𝑦))². Hence, inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐵(𝑧)}≥𝜇_𝐵(𝑦). Also, we have sup_{𝑧∈𝑥+𝑦}𝜆_𝐴(𝑧)≤max𝜆_𝐴(𝑥),𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥+𝑦}−𝜆_𝐴(𝑧)≥min−𝜆_𝐴(𝑥),−𝜆_𝐴𝑦⇒1−inf_{𝑧∈𝑥+𝑦}−𝜆_𝐴(𝑧)≥1+min−𝜆_𝐴(𝑥),−𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥+𝑦}1−𝜆_𝐴(𝑧)≥min1−𝜆_𝐴(𝑥),1−𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥+𝑦}1−𝜆_𝐴(𝑧)²≥min1−𝜆_𝐴(𝑥),1−𝜆_𝐴𝑦²⇒inf_{𝑧∈𝑥+𝑦}1−𝜆_𝐴(𝑧)²≥min1−𝜆_𝐴(𝑥)²,1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}−1−𝜆_𝐴(𝑧)²≤max−1−𝜆_𝐴(𝑥)²,−1−𝜆_𝐴𝑦²⇒1+sup_{𝑧∈𝑥+𝑦}−1−𝜆_𝐴(𝑧)²≤1+max−1−𝜆_𝐴(𝑥)²,−1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}1−1−𝜆_𝐴(𝑧)²≤max1−1−𝜆_𝐴(𝑥)²,1−1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}𝜆_𝐵(𝑧)≤max𝜆_𝐵(𝑥),𝜆_𝐵𝑦. (24) Moreover, we have sup_{𝑧∈𝑥𝛾𝑦}𝜆_𝐴(𝑧)≤𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥𝛾𝑦}−𝜆_𝐴(𝑧)≥−𝜆_𝐴𝑦⇒1−inf_{𝑧∈𝑥𝛾𝑦}−𝜆_𝐴(𝑧)≥1−𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥𝛾𝑦}1−𝜆_𝐴(𝑧)≥1−𝜆_𝐴𝑦⇒inf_{𝑧∈𝑥𝛾𝑦}1−𝜆_𝐴(𝑧)²≥1−𝜆_𝐴𝑦²⇒inf_{𝑧∈𝑥𝛾𝑦}1−𝜆_𝐴(𝑧)²≥1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}−1−𝜆_𝐴(𝑧)²≤−1−𝜆_𝐴𝑦²⇒1+sup_{𝑧∈𝑥+𝑦}−1−𝜆_𝐴(𝑧)²≤1−1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}1−1−𝜆_𝐴(𝑧)²≤1−1−𝜆_𝐴𝑦²⇒sup_{𝑧∈𝑥+𝑦}𝜆_𝐵(𝑧)≤𝜆_𝐵𝑦. (25)
We proved the theorem for left intuitionistic fuzzy Γ-hyperideals. For the proof of right intuitionistic fuzzy Γ-hyperideals similar proof is used.

Lemma 14. If {𝐴_𝑖=(𝜇_𝐴𝑖,𝜆_𝐴𝑖)} is a collection of intuitionistic fuzzy Γ-hyperideals of 𝑆, then ∩_𝑖∈Λ𝐴_𝑖 and ∪_𝑖∈Λ𝐴_𝑖 are intuitionistic fuzzy Γ-hyperideals of 𝑆, too.

Proof. The proof is straightforward.

Theorem 15. An intuitionistic fuzzy subset 𝐴=(𝜇_𝐴,𝜆_𝐴) of a Γ-semihyperring 𝑆 is a left (res. right) intuitionistic fuzzy Γ-hyperideal of 𝑆 if and only if for every 𝑡∈[0,1], the fuzzy sets 𝜇_𝐴 and 𝜆^𝑐_𝐴 are left (res. right) fuzzy Γ-hyperideals of 𝑆.

Proof. Assume that 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆. Clearly, by using the definition, we have that 𝜇_𝐴 is a left (res. right) fuzzy Γ-hyperideal.
Also, inf_{𝑧∈𝑥+𝑦}𝜆^𝑐_𝐴(𝑧)=inf_{𝑧∈𝑥+𝑦}1−𝜆_𝐴(𝑧)=1−sup_{𝑧∈𝑥+𝑦}𝜆_𝐴(𝑧)≥min1−𝜆_𝐴(𝑥),1−𝜆_𝐴𝑦=min𝜆^𝑐_𝐴(𝑥),𝜆^𝑐_𝐴𝑦,inf_{𝑧∈𝑥𝛾𝑦}𝜆^𝑐_𝐴(𝑧)=inf_{𝑧∈𝑥𝛾𝑦}1−𝜆_𝐴(𝑧)=1−sup_{𝑧∈𝑥𝛾𝑦}𝜆_𝐴(𝑧)≥1−𝜆_𝐴𝑦=𝜆^𝑐_𝐴𝑦, (26) for all 𝑥,𝑦∈𝑆 and 𝛾∈Γ.
Conversely, suppose that 𝜇_𝐴 and 𝜆^𝑐_𝐴 are left (res. right) fuzzy Γ-hyperideals of 𝑆. We have min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}≤inf_{𝑧∈𝑥+𝑦}{𝜇_𝐴(𝑧)} and 𝜇_𝐴(𝑦)≤inf_{𝑧∈𝑥𝛾𝑦}{𝜇_𝐴(𝑧)}, for all 𝑥,𝑦∈𝑆 and 𝛾∈Γ. We obtain sup_{𝑧∈𝑥+𝑦}𝜆_𝐴(𝑧)=sup_{𝑧∈𝑥+𝑦}1−𝜆^𝑐_𝐴(𝑧)=1−inf_{𝑧∈𝑥+𝑦}𝜆^𝑐_𝐴(𝑧)≤max1−𝜆^𝑐_𝐴(𝑥),1−𝜆^𝑐_𝐴𝑦=max𝜆_𝐴(𝑥),𝜆_𝐴𝑦,sup_{𝑧∈𝑥𝛾𝑦}𝜆_𝐴(𝑧)=sup_{𝑧∈𝑥𝛾𝑦}1−𝜆^𝑐_𝐴(𝑧)=1−inf_{𝑧∈𝑥𝛾𝑦}𝜆^𝑐_𝐴(𝑧)≤1−𝜆^𝑐_𝐴𝑦=𝜆_𝐴𝑦. (27) Therefore, 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆.
For any fuzzy set 𝜇 of 𝑆 and any 𝑡∈[0,1], 𝑈(𝜇;𝑡)={𝑥∈𝑆∣𝜇(𝑥)≥𝑡} is called an upper bound 𝑡-𝑙𝑒𝑣𝑒𝑙𝑐𝑢𝑡of 𝜇, and 𝐿(𝜇;𝑡)={𝑥∈𝑆∣𝜇(𝑥)≤𝑡} is called a lower bound 𝑡-𝑙𝑒𝑣𝑒𝑙𝑐𝑢𝑡of 𝜇.

Theorem 16. An intuitionistic fuzzy subset 𝐴 of a Γ-semihyperring 𝑆 is a left (right) intuitionistic fuzzy Γ-hyperideal of 𝑆 if and only if for every 𝑡,𝑠∈[0,1], the subsets 𝑈(𝜇_𝐴;𝑡) and 𝐿(𝜆_𝐴;𝑠) of 𝑆 are left (res. right) Γ-hyperideals, when they are non-empty.

Proof. Assume that 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆. Let 𝑥,𝑦∈𝑈(𝜇_𝐴;𝑡). Since inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧)≥min{𝜇_𝐴(𝑥),𝜇_𝐴(𝑦)}≥𝑡, then we have 𝑥+𝑦⊆𝑈(𝜇_𝐴;𝑡). Let 𝑥∈𝑆 and 𝑦∈𝑈(𝜇_𝐴;𝑡). We have inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝐴(𝑧)≥𝜇_𝐴(𝑦)≥𝑡. Therefore, 𝑧∈𝑈(𝜇_𝐴;𝑡) and so 𝑥𝛾𝑦⊆𝑈(𝜇_𝐴;𝑡).
Now, let 𝑥,𝑦∈𝐿(𝜆_𝐴;𝑠). Since 𝜆_𝐴(𝑥)≤𝑠 and 𝜆_𝐴(𝑦)≤𝑠, we have sup_{𝑧∈𝑥+𝑦}{𝜆_𝐴(𝑧)}≤max{𝜆_𝐴(𝑥),𝜆_𝐴(𝑦)}≤𝑠. Then, 𝑧∈𝐿(𝜆_𝐴;𝑠), and so 𝑥+𝑦⊆𝐿(𝜆_𝐴;𝑠). Now, for all 𝑥∈𝑆, 𝑦∈𝐿(𝜆_𝐴;𝑠) and 𝛾∈Γ we have sup_{𝑧∈𝑥𝛾𝑦}{𝜆_𝐴(𝑧)}≤𝜆_𝐴(𝑦)≤𝑠. Therefore, 𝑧∈𝐿(𝜆_𝐴;𝑠) and so 𝑥𝛾𝑦⊆𝐿(𝜆_𝐴;𝑠).
Conversely, suppose that all non-empty level sets  𝑈(𝜇_𝐴;𝑡) and 𝐿(𝜆_𝐴;𝑠) are left Γ-hyperideals. Let 𝑥,𝑦∈𝑆 and 𝛾∈Γ. Let 𝑡₀=𝜇_𝐴(𝑥), 𝑡₁=𝜇_𝐴(𝑦) and 𝑠₀=𝜆_𝐴(𝑥), 𝑠₁=𝜆_𝐴(𝑦) with 𝑡₀≤𝑡₁, 𝑠₀≤𝑠₁, then 𝑥,𝑦∈𝑈(𝜇_𝐴;𝑡₀) and 𝑥,𝑦∈𝐿(𝜆_𝐴;𝑠₁). Since 𝑥+𝑦⊆𝑈𝜇_𝐴;𝑡₀,𝑥+𝑦⊆𝐿𝜆_𝐴;𝑠₁, (28) we have 𝑡₀=min𝜇_𝐴(𝑥),𝜇_𝐴𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧,𝑠₁)=max𝜆_𝐴(𝑥),𝜆_𝐴𝑦≥sup_{𝑧∈𝑥+𝑦}𝜆_𝐴(𝑧). (29) Since 𝑈(𝜇_𝐴;𝑡) and 𝐿(𝜆_𝐴;𝑡) are left Γ-hyperideals, and we have 𝑧∈𝑥𝛾𝑦⊆𝑈(𝜇_𝐴;𝑡) and 𝑧∈𝑥𝛾𝑦⊆𝑈(𝜆_𝐴;𝑠), we have 𝜇_𝐴(𝑧)≥𝑡₀ and 𝜆_𝐴(𝑧)≤𝑠₁. And so inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝐴(𝑧)≥𝜇_𝐴(𝑦) and sup_{𝑧∈𝑥𝛾𝑦}𝜆_𝐴(𝑧)≤𝜆_𝐴(𝑦). Hence, 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆.

Corollary 17. Let 𝐼 be a left (res. right) Γ-hyperideal of a Γ-semihyperring 𝑆. We define fuzzy sets 𝜇_𝐴 and 𝜆_𝐴 as follows: 𝜇_𝐴=𝑎₀if𝑎𝑥∈𝐼,₁if𝜆𝑥∈𝑆−𝐼,_𝐴=𝑏₀if𝑏𝑥∈𝐼,₁if𝑥∈𝑆−𝐼, (30) where 0≤𝑎₁<𝑎₀,0≤𝑏₀<𝑏₁ and 𝑎_𝑖+𝑏_𝑖≤1 for 𝑖=0,1. Then, 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆 and 𝑈(𝜇;𝑎₀)=𝐼=𝐿(𝜆;𝑏₀).

Now, we give the following theorem about the intuitionistic fuzzy sets □𝐴 and ⋄𝐴.

Theorem 18. If 𝐴=(𝜇_𝐴,𝜆_𝐴) is an intuitionistic fuzzy Γ-hyperideal of 𝑆, then so are □𝐴 and ⋄𝐴.

Proof. Since 𝐴=(𝜇_𝐴,𝜆_𝐴) is an intuitionistic fuzzy Γ-hyperideal of 𝑆, we have min𝜇_𝐴(𝑥),𝜇_𝐴𝑦≤inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧,𝜇_𝐴)(𝑥)≤inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝐴(𝑧), (31) for all 𝑥,𝑦,𝑧∈𝑆 and 𝛾∈Γ. Moreover, sup_{𝑧∈𝑥+𝑦}𝜇^𝑐_𝐴(𝑧)=sup_{𝑧∈𝑥+𝑦}1−𝜇_𝐴(𝑧)=1−inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧)≤max1−𝜇_𝐴(𝑥),1−𝜇_𝐴𝑦=max𝜇^𝑐_𝐴(𝑥),𝜇^𝑐_𝐴𝑦,sup_{𝑧∈𝑥𝛾𝑦}𝜇^𝑐_𝐴(𝑧)=sup_{𝑧∈𝑥𝛾𝑦}1−𝜇_𝐴(𝑧)=1−inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝐴(𝑧)≤max1−𝜇_𝐴(𝑥),1−𝜇_𝐴𝑦=max𝜇^𝑐_𝐴(𝑥),𝜇^𝑐_𝐴𝑦. (32) This shows that □𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆. For the other part one can use the similar way.
Note that if we have a fuzzy Γ-hyperideal 𝐴, then □𝐴=𝐴=⋄𝐴. While for a proper intuitionistic fuzzy Γ-hyperideal 𝐴, we have □𝐴⊂𝐴⊂⋄𝐴 and □𝐴≠𝐴≠⋄𝐴.

Theorem 19. If 𝐴=(𝜇_𝐴,𝜆_𝐴) is a left intuitionistic fuzzy Γ-hyperideal of 𝑆, then we have, 𝜇_𝐴(𝑥)=sup𝑡∈[0,1]∣𝑥∈𝑈𝜇_𝐴,𝜆;𝑡_𝐴(𝑥)=inf𝑠∈[0,1]∣𝑥∈𝐿𝜆_𝐴;𝑠, (33) for all 𝑥∈𝑆.

Proof. The proof is straightforward.

Let 𝐴=(𝜇_𝐴,𝜆_𝐴) and 𝐵=(𝜇_𝐵,𝜆_𝐵) be two intuitionistic fuzzy Γ-hyperideals of 𝑆. Then, the product of 𝐴 and 𝐵 denoted by 𝐴Γ𝐵 is defined as follows: 𝜇_𝐴,𝜆_𝐴Γ𝜇_𝐵,𝜆_𝐵(𝑧)=𝑡₁,𝑡₂, (34) where 𝑡₁=sup_{𝑧∈𝑥𝛾𝑦}min𝜃₁(𝑥),𝜎₁𝑦,0∣𝑥,𝑦∈𝑆,𝛾∈Γotherwise,𝑡₂=inf_{𝑧∈𝑥𝛾𝑦}max𝜃₂(𝑥),𝜎₂𝑦,0∣𝑥,𝑦∈𝑆,𝛾∈Γotherwise. (35)

Let 𝜒_𝐼 be the characteristic function of a left (res. right) Γ-hyperideal 𝐼 of 𝑆. Then, 𝐼=(𝜒_𝐼,𝜒_𝐼^𝐶) is a left (res. right) intuitionistic fuzzy Γ-hyperideal of 𝑆.

Theorem 20. If 𝑆 is a Γ-semihyperring and 𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆, then 𝑋Γ𝐴⊆𝐴, where 𝑋=(𝜒_𝑆,𝜒^𝑐_𝑆) and 𝜒_𝑆 is the characteristic function of 𝑆.

Proof. The proof is straightforward.

Proposition 21. Let 𝑆 be a Γ-semihyperring, and let 𝐴 be an intuitionistic fuzzy Γ-hyperideal of 𝑆. Then, (1) if 𝑤 is a fixed element of 𝑆, then the set 𝜇_𝐴^𝑤={𝑥∈𝑆∣𝜇(𝑥)≥𝜇(𝑤)} and 𝜆_𝐴^𝑤={𝑥∈𝑆∣𝜆(𝑥)≤𝜆(𝑤)} are Γ-hyperideals of 𝑆,(2) the sets 𝑈={𝑥∈𝑆∣𝜇_𝐴(𝑥)=0} and 𝐿={𝑥∈𝑆∣𝜆_𝐴(𝑥)=0} are Γ-hyperideals of 𝑆.

Proof. (1) Suppose that 𝑥,𝑦∈𝜇_𝐴^𝑤. Then, we have inf_{𝑧∈𝑥+𝑦}𝜇_𝐴(𝑧)≥min𝜇_𝐴(𝑥),𝜇_𝐴𝑦≥𝜇(𝑤), (36) which implies that 𝑧∈𝑈(𝜇_𝐴;𝜇(𝑤)) and so 𝑥+𝑦⊆𝑈(𝜇_𝐴;𝜇(𝑤)). Similarly, suppose that 𝑥,𝑦∈𝜆_𝐴^𝑤. We have sup_{𝑧∈𝑥+𝑦}𝜆_𝐴(𝑧)≤max{𝜆_𝐴(𝑥),𝜆_𝐴(𝑦)}≤𝜇(𝑤) which implies that 𝑧∈𝑈(𝜆_𝐴;𝜆(𝑤)) and so 𝑥+𝑦⊆𝑈(𝜆_𝐴;𝜆(𝑤)). Assume that 𝑥∈𝑆, 𝑦∈𝜇_𝐴^𝑤 and 𝛾∈Γ. Then, we have inf_{𝑧∈𝑥𝛾𝑦}𝜇_𝐴(𝑧)≥𝜇_𝐴(𝑦)≥𝜇(𝑤) which implies that 𝑧∈𝑈(𝜇_𝐴;𝜇(𝑤)) and so 𝑥𝛾𝑦⊆𝑈(𝜇_𝐴;𝜇(𝑤)). Similarly, suppose that 𝑥∈𝜆_𝐴^𝑤, 𝑦∈𝑆 and 𝛾∈Γ. We have sup_{𝑧∈𝑥𝛾𝑦}𝜆_𝐴(𝑧)≤𝜆_𝐴(𝑥)≤𝜇(𝑤) which implies that 𝑧∈𝑈(𝜆_𝐴;𝜆(𝑤)) and so 𝑥𝛾𝑦⊆𝑈(𝜆_𝐴;𝜆(𝑤)).
(2) One can easily prove the second part.

Definition 22. Let 𝑆 be a Γ-semihypering, and let 𝑆 be a Γ-semihyperring. If there exists a map 𝜑∶𝑆→𝑆 and a bijection 𝑓∶Γ→Γ, such that 𝜑(𝑥)𝑦,𝜑(𝑥)𝑓𝛾𝜑𝑦,𝑥+𝑦=𝜑+𝜑𝑥𝛾𝑦=𝜑 (37) for all 𝑥,𝑦∈𝑆 and 𝛾∈Γ, then we say (𝜑,𝑓) is a homomorphism from 𝑆 to 𝑆. Also, if 𝜑 is a bijection, then (𝜑,𝑓) is called an isomorphism, and 𝑆 and 𝑆 are isomorphic.

Definition 23. Let 𝐴=(𝜇_𝐴,𝜆_𝐴) and 𝐵=(𝜇_𝐵,𝜆_𝐵) be two non-empty intuitionistic fuzzy subsets of 𝑆 and 𝑆^_, respectively, and let 𝜑∶𝑆→𝑆 be a map. Then, (1)the inverse image of 𝐵 under 𝜑 is defined by 𝜑⁻¹(𝐵)=𝜑⁻¹𝜇_𝐵,𝜑⁻¹𝜆_𝐵, (38) where 𝜑⁻¹𝜇_𝐵=𝜇_𝐵∘𝜑,𝜑⁻¹𝜆_𝐵=𝜆_𝐵∘𝜑, (39)(2)the image of 𝐴 under 𝜑, denoted by 𝜑(𝐴), is the intuitionistic fuzzy set in 𝑆 defined by 𝜑(𝐴)=(𝜑(𝜇_𝐴),𝜑^∗(𝜆_𝐴)), where for each 𝑦∈𝑆𝜑𝜇_𝐴𝑦=sup_{𝑥∈𝜑−1𝑦}𝜇_𝐴(𝑥)if𝜑⁻¹𝑦0≠∅,otherwise,𝜑^∗𝜆_𝐴𝑦=inf_{𝑥∈𝜑−1𝑦}𝜆_𝐴(𝑥)if𝜑⁻¹𝑦1≠∅,otherwise. (40)

Now, the next theorem is about image and preimage of intuitionistic fuzzy sets.

Theorem 24. Let 𝑆 be a Γ-semihyperring, let 𝑆 be a Γ-semihyperring, and let (𝜑,𝑓) be a homomorphism from 𝑆 to 𝑆. Then,(1) if 𝐴=(𝜇_𝐴,𝜆_𝐴) is an intuitionistic fuzzy Γ-hyperideal of 𝑆, then 𝜑(𝐴) is an intuitionistic fuzzy Γ-hyperideal of𝑆, (2) if 𝐵=(𝜇_𝐵,𝜆_𝐵) is an intuitionistic fuzzy Γ-hyperideal of𝑆, then 𝜑⁻¹(𝐵) is an intuitionistic fuzzy Γ-hyperideal of𝑆.

Proof. The proof is straightforward.

4. Artinian and Noetherian Γ-Semihypergroups

In this section, we give some characterizations of Artinian and Noetherian Γ-semihyperrings.

Definition 25. Let 𝑆 be a Γ-semihypering. Then, 𝑆 is called Noetherian (Artinian resp.) if 𝑆 satisfies the ascending (descending) chain condition on Γ-hyperideals. That is, for any Γ-hyperideals of 𝑆, such that 𝐼₁⊆𝐼₂⊆𝐼₃⊆⋯⊆𝐼_𝑖⋯𝐼₁⊇𝐼₂⊇𝐼₃⊇⋯⊇𝐼_𝑖⋯, (41) there exists 𝑛∈ℕ, such that 𝐼_𝑖=𝐼_𝑖+1, for all 𝑖≥𝑛.

Proposition 26 (see [28]). Let 𝑆 be a Γ-semihypering. Then, the following conditions are equivalent:(1)𝑆 is Noetherian.(2)𝑆 satisfies the maximum condition for Γ-hyperideals.(3) Every Γ-hyperideal of 𝑆 is finitely generated.

Theorem 27. If every intuitionistic fuzzy Γ-hyperideal of Γ-semihypering has finite number of values, then 𝑆 is Artinian.

Proof. Suppose that every intuitionistic fuzzy Γ-hyperideal of Γ-semihyperings has finite number of values, and 𝑆 is not Artinian. So, there exists a strictly descending chain 𝑆=𝐼₀⊇𝐼₁⊇𝐼₂⊇⋯ (42) of Γ-hyperideals of 𝑆. We define the intuitionistic fuzzy set 𝐴 by 𝜇_𝐴(𝑥)=𝑛𝑛+1if𝑥∈𝐼_𝑛−𝐼_𝑛+1,1if𝑥∈⋂^∞_𝑛=0𝐼_𝑛,𝜆_𝐴(𝑥)=1𝑛+1if𝑥∈𝐼_𝑛−𝐼_𝑛+1,0if𝑥∈⋂^∞_𝑛=0𝐼_𝑛. (43) It is easy to see that 𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆. We have contradiction because of the definition of 𝐴, which depends on 𝐼₀⊇𝐼₁⊇𝐼₂⊇⋯ infinitely descending chain of Γ-hyperideals of 𝑆.

Theorem 28. Let 𝑆 be a Γ-semihypering. Then the following statements are equivalent:(1)𝑆 is Noetherian.(2) The set of values of any intuitionistic fuzzy Γ-hyperideal of 𝑆 is a well-ordered subset of [0,1].

Proof. (1)⇒(2) Let 𝐴 be an intuitionistic fuzzy Γ-hyperideal of Γ-semihypering. Assume that the set of values of 𝐴 is not a well-ordered subset of [0,1]. Then, there exits a strictly infinite decreasing sequence {𝑡_𝑛}, such that 𝜇_𝐴(𝑥)=𝑡_𝑛 and 𝜆_𝐴(𝑥)=1−𝑡_𝑛. Let 𝐼_𝑛={𝑥∈𝑆∣𝜇(𝑥)≥𝑡_𝑛} and 𝐽_𝑛={𝑥∈𝑆∣𝜆(𝑥)≤1−𝑡_𝑛}. Then, 𝐼₁⊆𝐼₂⊆⋯ and 𝐽₁⊆𝐽₂⊆⋯ are strictly infinite decreasing chains of Γ-hyperideals of 𝑆. This contradicts with our hypothesis.
(2)⇒(1) Suppose that 𝐼₁⊆𝐼₂⊆⋯ is a strictly infinite ascending chain Γ-hyperideals of 𝑆. Let 𝐼=⋃_𝑛∈ℕ𝐼_𝑛. It is easy to see that 𝐼 is a Γ-hyperideal of 𝑆. Now, we define 𝜇_𝐴(𝑥)=0if1𝑥∉𝐼,𝑘where𝑘=min𝑛∈ℕ∣𝑥∈𝐼_𝑛,𝜆_𝐴(𝑥)=1if𝑥∉𝐼,𝑘−1𝑘+1where𝑘=max𝑛∈ℕ∣𝑥∈𝐼_𝑛. (44) Clearly, 𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆. Since the chain is not finite, 𝐴 has strictly infinite ascending sequence of values. This contradicts that the set of the values of fuzzy Γ-hyperideal is not well ordered.

Theorem 29. If a Γ-semihyperring 𝑆 both Artinian and Noetherian, then every intuitionistic fuzzy Γ-hyperideal of 𝑆 is finite valued.

Proof. Suppose that 𝐴 is an intuitionistic fuzzy Γ-hyperideal of 𝑆, Im(𝜇_𝐴) and Im(𝜆_𝐴) are not finite. According the previous theorem, we consider the following two cases.
Case 1. Suppose that 𝑡₁<𝑡₂<𝑡₃<⋯ is strictly increasing sequence in Im(𝜇_𝐴) and 𝑠₁>𝑠₂>𝑠₃⋯ is strictly decreasing sequence in Im(𝜆_𝐴). Now, we obtain 𝑈𝜇_𝐴;𝑡₁⊃𝑈𝜇_𝐴;𝑡₂⊃𝑈𝜇_𝐴;𝑡₃𝐿𝜆⊃⋯,_𝐴;𝑠₁⊃𝐿𝜆_𝐴;𝑠₂⊃𝐿𝜆_𝐴;𝑠₃⊃⋯ (45) are strictly descending and ascending chains ofΓ-hyperideals, respectively. Since 𝑆 is both Artinian and Noetherian there exists a natural number 𝑖, such that 𝑈(𝜇_𝐴;𝑡_𝑖)=𝑈(𝜇_𝐴;𝑡_𝑖+𝑛) and 𝐿(𝜆_𝐴;𝑠_𝑖)=𝐿(𝜆_𝐴;𝑠_𝑖+𝑛) for 𝑛≥1. This means that 𝑡_𝑖=𝑡_𝑖+𝑛 and 𝑠_𝑖=𝑠_𝑖+𝑛. This is a contradiction.
Case 2. Assume that 𝑡₁>𝑡₂>𝑡₃>⋯, 𝑠₁<𝑠₂<𝑠₃⋯ are strictly decreasing and increasing sequence in Im(𝜇_𝐴) and Im(𝜆_𝐴), respectively. From these equalities we conclude that 𝑈𝜇_𝐴;𝑡₁⊂𝑈𝜇_𝐴;𝑡₂⊂𝑈𝜇_𝐴;𝑡₃𝐿𝜆⊂⋯,_𝐴;𝑠₁⊂𝐿𝜆_𝐴;𝑠₂⊂𝐿𝜆_𝐴;𝑠₃⊂⋯ (46) are strictly ascending and descending chains of Γ-hyperideals, respectively. Because of the hypothesis there exists a natural number 𝑗, such that 𝑈(𝜇_𝐴;𝑡_𝑖)=𝑈(𝜇_𝐴;𝑡_𝑖+𝑛) and 𝐿(𝜆_𝐴;𝑠_𝑖)=𝐿(𝜆_𝐴;𝑠_𝑖+𝑛) for 𝑛≥1. This implies that 𝑡_𝑖=𝑡_𝑖+𝑛 and 𝑠_𝑖=𝑠_𝑖+𝑛 which is a contradiction.

Acknowledgments

This research is supported by TUBITAK-BIDEB. The paper was essentially prepared during the second author’s stay at the Department of Mathematics, Yildiz Technical University in 2011. The second author is greatly indebted to Dr. B. A. Ersoy for his hospitality.