# Structure of Intuitionistic Fuzzy Sets inΓ-Semihyperrings

Authors: Bayram Ali Ersoy; Bijan Davvaz
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 introducedΓ-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

LetS 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
(1)A∘B=∪a∈A,b∈Ba∘b,x∘A={x}∘A,A∘x=A∘{x}.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,
(2)∪u∈x∘yu∘z=∪v∈y∘zx∘v.
A semihypergroup (S,∘) is called a hypergroup if for all x∈S, x∘S=S∘x=S.Asemihyperringis 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,(3)x·(y+z)=x·y+x·z,(x+y)·z=x·z+y·z,
for all x,y,z∈S.(4)
The element0∈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 semihyperringS has identity element if there exists 1S∈S, such that 1S·x=x·1S=x, for all x∈S. An element x∈S is calledunitif there exists y∈S, such that y·x=x·y=1S. A non-empty subset A of a semihyperring (S,+,·) is called subsemihyperringif 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.LetS 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
(4)(xβy)γz=xβ(yγz).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 ofΓ-semihyperringwas introduced and studied by Dehkordi and Davvaz [26–28]. We recall the following definition from [26].Definition 1.
LetS be a commutative semihypergroup, and Γ be a commutative group. Then, S is called a Γ-semihyperringif 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 allx,y,z∈S and α,β∈Γ.In the above definition, ifS is a semigroup, then S is called a multiplicative Γ-semihyperring. A Γ-semihyperringS is called commutativeif 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-semihyperringand x∈S. We define
(5)A+B={x∣x∈a+b,a∈A,b∈B},AΓB={x∣x∈aαb,a∈A,b∈B,α∈Γ}.A non-empty subset ofS1 of Γ-semihyperring S is called a sub-Γ-semihyperringif 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
(6)S1+S1⊆S1,S1ΓS1⊆S1.Example 2.
Let(S,+,·) be a semiring, and let Γ be a subsemiring of S. We define
(7)xγy=〈x,γ,y〉,
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.
LetS=ℚ+, Γ={γi∣i∈ℕ} and Ai=iℤ+. We define xγiy→xAiy 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 Mm,n(S) the set of all m×n matrices with entries in S. We define ∘:Mm,n(S)×Mn,m(S)×Mm,n(S)→P*(Mm,n(S)) by
(8)A∘B∘C={Z∈Mm,n(S)∣Z∈ABC},
for all A,C∈Mm,n(S) and B∈Mn,m(S).
Then,Mm,n(S) is Mn,m(S)-semihyperring [29]. Now, suppose that
(9)X={(aij)m,n∣a11≠0,a21≠0,aij=0otherwise}.
Now, it is easy to see that X is a sub Γ-semihyperring of S.
A non-empty subsetI of a Γ-semihyperring S is a left(right) Γ-hyperidealof S if for any I1,I2⊆I implies I1+I2⊆I and IΓS⊆I(SΓI⊆I) and is a Γ-hyperidealof S if it is both left and right Γ-hyperideal.Example 5.
Consider the following:(10)S={[abcd]∣a,b,c,d∈ℤ+∪{0}},Ai={[ie00if]∣e,f∈2ℤ+},∀i∈ℕ,Γ={γi∣i∈ℕ}.
Then, S is a Γ-semihyperring under the matrix addition and the hyperoperation as follows:
(11)xγiy=xAiy,
for all x,y∈S and γi∈Γ. Let
(12)I1={[ab00]∣a,b∈ℤ+∪{0}},I2={[a0c0]∣a,c∈ℤ+∪{0}},I3={[abcd]∣a,b,c,d∈2ℤ+∪{0}}.
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 subsetμ of a non-empty set S is a function from S to [0,1]. For all x∈Sμc, the complement of μ is the fuzzy subset defined by μc(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
(13)(μ∩λ)(x)=min{μ(x),λ(x)},(μ∪λ)(x)=max{μ(x),λ(x)}.Definition 6.
LetS be a Γ-semihyperring, and let μ be a fuzzy subset of S. Then,(1)
μ is called a fuzzy leftΓ-hyperidealof S if
(14)min{μ(x),μ(y)}≤infz∈x+y{μ(z)},μ(y)≤infz∈xγy{μ(z)}∀x,y∈S,∀γ∈Γ.(2)
μ is called a fuzzy rightΓ-hyperidealof S if
(15)min{μ(x),μ(y)}≤infz∈x+y{μ(z)},μ(x)≤infz∈xγy{μ(z)}∀x,y∈S,∀γ∈Γ.(3)
μ is called a fuzzyΓ-hyperidealof S if μ is both a fuzzy left Γ-hyperideal and fuzzy right Γ-hyperideal of S.Example 7.
LetS1=ℤ6, Γ={γ2,γ3}, S2={0-,3-}, and S3={0-} be non-empty subsets of S1. We define xγiy=xSiy, for every γi∈Γ and x,y∈S1. Then, S1 is a Γ-semihypering. Now, we define the fuzzy subset μ of S1 as follows:
(16)μ(x)={0ifx=1-,2-,4-,5-,13ifx=3-,1ifx=0-.
It is easy to see that μ is a fuzzy Γ-hyperideal of S1.


## 3. Atanassov’s Intuitionistic FuzzyΓ-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,μA(x),λA(x))∣x∈S}. Here, μA:S→[0,1] is the degree of membership of the element x∈S to the set A, and λA:S→[0,1] is the degree of nonmembership of the element x∈S to the set A. We have also 0≤μA(x)+λA(x)≤1, for all x∈S.Example 8 (see [35]).
Consider the universeX={10,100,500,1000,1200}. An intuitionistic fuzzy set “Large” of X denoted by A and may be defined by
(17)A={〈10,0.01,0.9〉,〈100,0.1,0.88〉,〈500,0.4,0.05〉,〈1000,0.8,0.1〉,〈1200,1,0〉}.
One may define an intuitionistic fuzzy set “Very Large” denoted byB, as follows:
(18)μB(x)=(μA(x))2,λB(x)=1-(1-λA(x))2,
for all x∈X. Then,
(19)B={〈10,0.0001,0.99〉,〈100,0.01,0.9856〉,〈500,0.16,0.75〉,〈1000,0.64,0.19〉,〈1200,1,0〉}.
For the sake of simplicity, we shall use the symbol A=(μA,λA) instead of A={(x,μA(x),λA(x))∣x∈S}. Let A=(μA,λA) and B=(μB,λB) be two intuitionistic fuzzy sets of S. Then, the following expressions are defined in [4] as follows:(1)
A⊆B if and only if μA(x)≤μB(x) and λA(x)≥λB(x), for all x∈S,(2)
AC={(x,λA(x),μA(x))∣x∈S},(3)
A∩B={(x,min{μA(x),μB(x)},max{λA(x),λB(x)})∣x∈S},(4)
A∪B={(x,max{μA(x),μB(x)},min{λA(x),λB(x)})∣x∈S},(5)
□A={(x,μA(x),μAc(x))∣x∈S},(6)
⋄A={(x,λAc(x),λA(x))∣x∈S}.
Now, we introduce the notion of intuitionistic fuzzyΓ-hyperideals of Γ-semihyperrings.Definition 9.
LetS be a Γ-semihyperring. (1)
An intuitionistic fuzzy setA=(μA,λA) in S is called a left intuitionistic fuzzyΓ-hyperideal of Γ-semihyperring S if
(i)
min{μA(x),μA(y)}≤infz∈x+y{μA(z)}  and max{λA(x),λA(y)}≥supz∈x+y{λA(z)},(ii)
μA(y)≤infz∈xγy{μA(z)} and supz∈xγy{λA(z)}≤λA(y),for allx,y∈S,γ∈Γ.(2)
An intuitionistic fuzzy setA=(μA,λA) in S is called a right intuitionistic fuzzyΓ-hyperideal of Γ-semihyperring S if
(i)
min{μA(x),μA(y)}≤infz∈x+y{μA(z)}  and max{λA(x),λA(y)}≥supz∈x+y{λA(z)},(ii)
μA(x)≤infz∈xγy{μA(z)}  and supz∈xγy{λA(z)}≤λA(x),for allx,y∈S,γ∈Γ.(3)
An intuitionistic fuzzy setA=(μA,λA) in S is called an intuitionistic fuzzyΓ-hyperideal of Γ-semihyperring S if it is both left intuitionistic fuzzy Γ-hyperideal and right intuitionistic fuzzy Γ-hyperideal of S.Example 10.
LetS1 be a Γ-semihyperring in Example 7. We define the fuzzy subsets μA and λA of S1 as follows:
(20)μ(x)={0ifx=1-,2-,4-,5-,14ifx=3-,25ifx=0-,λ(x)={12ifx=1-,2-,4-,5-,13ifx=3-,310ifx=0-.
Then, A is an intuitionistic fuzzy Γ-hyperideal of S1.Example 11.
LetS be a Γ-semihyperring defined in Example 5. Suppose that
(21)μAi(x)={58ifa,b,c,d∈2ℤ+∪{0},25otherwise,λAi(x)={19ifa,b,c,d∈2ℤ+∪{0},47otherwise,
for all x∈S. Then, Ai=(μAi,λAi) is an intuitionistic fuzzy Γ-hyperideal of S.Theorem 12.
LetA=(μA,λA) be an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring S and 0≤t≤1. We define an intuitionistic fuzzy set B=(μB,λB) in S by μB(x)=tμA(x) and λB(x)=(1-t)λB(x), for all x∈S. Then, B=(μB,λB) is an intuitionistic fuzzy Γ-hyperideal of S.Proof.
Note that0≤μB(x)+λB(x)=tμA(x)+(1-t)λA(x)≤t+1-t=1.Theorem 13.
LetA=(μA,λA) be an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring S. We define an intuitionistic fuzzy set B=(μB,λB) in S by μB(x)=(μA(x))2 and λB(x)=1-(1-λB(x))2, for all x∈S. Then, B=(μB,λB) is an intuitionistic fuzzy Γ-hyperideals of S.Proof.
SinceA=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal, we have infz∈x+y{μA(z)}≥min{μA(x),μA(y)} and so
(22)(infz∈x+y{μA(z)})2≥(min{μA(x),μA(y)})2.
Hence, infz∈x+y{(μA(z))2}≥min{(μA(x))2,(μA(y))2}; that is,
(23)infz∈x+y{μB(z)}≥min{μB(x),μB(y)}.
Since infz∈xγy{μA(z)}≥μA(y), so (infz∈xγy{μA(z)})2≥(μA(y))2 which implies that infz∈xγy{(μA(z))2}≥(μA(y))2. Hence, infz∈xγy{μB(z)}≥μB(y). Also, we have
(24)supz∈x+y{λA(z)}≤max{λA(x),λA(y)}⇒infz∈x+y{-λA(z)}≥min{-λA(x),-λA(y)}⇒1-infz∈x+y{-λA(z)}≥1+min{-λA(x),-λA(y)}⇒infz∈x+y{1-λA(z)}≥min{1-λA(x),1-λA(y)}⇒(infz∈x+y{1-λA(z)})2≥(min{1-λA(x),1-λA(y)})2⇒infz∈x+y{(1-λA(z))2}≥min{(1-λA(x))2,(1-λA(y))2}⇒supz∈x+y{-(1-λA(z))2}≤max{-(1-λA(x))2,-(1-λA(y))2}⇒1+supz∈x+y{-(1-λA(z))2}≤1+max{-(1-λA(x))2,-(1-λA(y))2}⇒supz∈x+y{1-(1-λA(z))2}≤max{1-(1-λA(x))2,1-(1-λA(y))2}⇒supz∈x+y{λB(z)}≤max{λB(x),λB(y)}.
Moreover, we have
(25)supz∈xγy{λA(z)}≤λA(y)⇒infz∈xγy{-λA(z)}≥-λA(y)⇒1-infz∈xγy{-λA(z)}≥1-λA(y)⇒infz∈xγy{1-λA(z)}≥1-λA(y)⇒(infz∈xγy{1-λA(z)})2≥(1-λA(y))2⇒infz∈xγy{(1-λA(z))2}≥(1-λA(y))2⇒supz∈x+y{-(1-λA(z))2}≤-(1-λA(y))2⇒1+supz∈x+y{-(1-λA(z))2}≤1-(1-λA(y))2⇒supz∈x+y{1-(1-λA(z))2}≤1-(1-λA(y))2⇒supz∈x+y{λB(z)}≤λB(y).
We proved the theorem for left intuitionistic fuzzyΓ-hyperideals. For the proof of right intuitionistic fuzzy Γ-hyperideals similar proof is used.Lemma 14.
If{Ai=(μAi,λAi)} is a collection of intuitionistic fuzzy Γ-hyperideals of S, then ∩i∈ΛAi and ∪i∈ΛAi are intuitionistic fuzzy Γ-hyperideals of S, too.Proof.
The proof is straightforward.Theorem 15.
An intuitionistic fuzzy subsetA=(μA,λA) 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 μA and λAc are left (res. right) fuzzy Γ-hyperideals of S.Proof.
Assume thatA=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S. Clearly, by using the definition, we have that μA is a left (res. right) fuzzy Γ-hyperideal.
Also,(26)infz∈x+y{λAc(z)}=infz∈x+y{1-λA(z)}=1-supz∈x+y{λA(z)}≥min{1-λA(x),1-λA(y)}=min{λAc(x),λAc(y)},infz∈xγy{λAc(z)}=infz∈xγy{1-λA(z)}=1-supz∈xγy{λA(z)}≥1-λA(y)=λAc(y),
for all x,y∈S and γ∈Γ.
Conversely, suppose thatμA and λAc are left (res. right) fuzzy Γ-hyperideals of S. We have min{μA(x),μA(y)}≤infz∈x+y{μA(z)} and μA(y)≤infz∈xγy{μA(z)}, for all x,y∈S and γ∈Γ. We obtain
(27)supz∈x+y{λA(z)}=supz∈x+y{1-λAc(z)}=1-infz∈x+y{λAc(z)}≤max{1-λAc(x),1-λAc(y)}=max{λA(x),λA(y)},supz∈xγy{λA(z)}=supz∈xγy{1-λAc(z)}=1-infz∈xγy{λAc(z)}≤1-λAc(y)=λA(y).
Therefore, A=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S.
For any fuzzy setμ of S and any t∈[0,1], U(μ;t)={x∈S∣μ(x)≥t} is called an upper boundt-levelcutof μ, and L(μ;t)={x∈S∣μ(x)≤t} is called a lower boundt-levelcutof μ.Theorem 16.
An intuitionistic fuzzy subsetA 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(μA;t) and L(λA;s) of S are left (res. right) Γ-hyperideals, when they are non-empty.Proof.
Assume thatA=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S. Let x,y∈U(μA;t). Since infz∈x+yμA(z)≥min{μA(x),μA(y)}≥t, then we have x+y⊆U(μA;t). Let x∈S and y∈U(μA;t). We have infz∈xγyμA(z)≥μA(y)≥t. Therefore, z∈U(μA;t) and so xγy⊆U(μA;t).
Now, letx,y∈L(λA;s). Since λA(x)≤s and λA(y)≤s, we have supz∈x+y{λA(z)}≤max{λA(x),λA(y)}≤s. Then, z∈L(λA;s), and so x+y⊆L(λA;s). Now, for all x∈S, y∈L(λA;s) and γ∈Γ we have supz∈xγy{λA(z)}≤λA(y)≤s. Therefore, z∈L(λA;s) and so xγy⊆L(λA;s).
Conversely, suppose that all non-empty level setsU(μA;t) and L(λA;s) are left Γ-hyperideals. Let x,y∈S and γ∈Γ. Let t0=μA(x), t1=μA(y) and s0=λA(x), s1=λA(y) with t0≤t1, s0≤s1, then x,y∈U(μA;t0) and x,y∈L(λA;s1). Since
(28)x+y⊆U(μA;t0),x+y⊆L(λA;s1),
we have
(29)t0=min{μA(x),μA(y)}≤infz∈x+y{μA(z)},s1=max{λA(x),λA(y)}≥supz∈x+y{λA(z)}.
Since U(μA;t) and L(λA;t) are left Γ-hyperideals, and we have z∈xγy⊆U(μA;t) and z∈xγy⊆U(λA;s), we have μA(z)≥t0 and λA(z)≤s1. And so infz∈xγyμA(z)≥μA(y) and supz∈xγyλA(z)≤λA(y). Hence, A=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S.Corollary 17.
LetI be a left (res. right) Γ-hyperideal of a Γ-semihyperring S. We define fuzzy sets μA and λA as follows:
(30)μA={a0ifx∈I,a1ifx∈S-I,λA={b0ifx∈I,b1ifx∈S-I,
where 0≤a1<a0,0≤b0<b1 and ai+bi≤1 for i=0,1. Then, A=(μA,λA) 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.
IfA=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, then so are □A and ⋄A.Proof.
SinceA=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, we have
(31)min{μA(x),μA(y)}≤infz∈x+y{μA(z)},μA(x)≤infz∈xγy{μA(z)},
for all x,y,z∈S and γ∈Γ. Moreover,
(32)supz∈x+y{μAc(z)}=supz∈x+y{1-μA(z)}=1-infz∈x+yμA(z)≤max{1-μA(x),1-μA(y)}=max{μAc(x),μAc(y)},supz∈xγy{μAc(z)}=supz∈xγy{1-μA(z)}=1-infz∈xγyμA(z)≤max{1-μA(x),1-μA(y)}=max{μAc(x),μAc(y)}.
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 fuzzyΓ-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.
IfA=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S, then we have,
(33)μA(x)=sup{t∈[0,1]∣x∈U(μA;t)},λA(x)=inf{s∈[0,1]∣x∈L(λA;s)},
for all x∈S.Proof.
The proof is straightforward.LetA=(μA,λA) and B=(μB,λB) be two intuitionistic fuzzy Γ-hyperideals of S. Then, the product of A and B denoted by AΓB is defined as follows:
(34)(μA,λA)Γ(μB,λB)(z)=(t1,t2),
where
(35)t1={supz∈xγy{min(θ1(x),σ1(y))∣x,y∈S,γ∈Γ},0otherwise,t2={infz∈xγy{max(θ2(x),σ2(y))∣x,y∈S,γ∈Γ},0otherwise.LetχI be the characteristic function of a left (res. right) Γ-hyperideal I of S. Then, I=(χI,χIC) is a left (res. right) intuitionistic fuzzy Γ-hyperideal of S.Theorem 20.
IfS is a Γ-semihyperring and A is an intuitionistic fuzzy Γ-hyperideal of S, then XΓA⊆A, where X=(χS,χSc) and χS is the characteristic function of S.Proof.
The proof is straightforward.Proposition 21.
LetS be a Γ-semihyperring, and let A be an intuitionistic fuzzy Γ-hyperideal of S. Then, (1)
ifw is a fixed element of S, then the set μAw={x∈S∣μ(x)≥μ(w)} and λAw={x∈S∣λ(x)≤λ(w)} are Γ-hyperideals of S,(2)
the setsU={x∈S∣μA(x)=0} and L={x∈S∣λA(x)=0} are Γ-hyperideals of S.Proof.
(1) Suppose thatx,y∈μAw. Then, we have
(36)infz∈x+yμA(z)≥min{μA(x),μA(y)}≥μ(w),
which implies that z∈U(μA;μ(w)) and so x+y⊆U(μA;μ(w)). Similarly, suppose that x,y∈λAw. We have supz∈x+yλA(z)≤max{λA(x),λA(y)}≤μ(w) which implies that z∈U(λA;λ(w)) and so x+y⊆U(λA;λ(w)). Assume that x∈S, y∈μAw and γ∈Γ. Then, we have infz∈xγyμA(z)≥μA(y)≥μ(w) which implies that z∈U(μA;μ(w)) and so xγy⊆U(μA;μ(w)). Similarly, suppose that x∈λAw, y∈S and γ∈Γ. We have supz∈xγyλA(z)≤λA(x)≤μ(w) which implies that z∈U(λA;λ(w)) and so xγy⊆U(λA;λ(w)).
(2) One can easily prove the second part.Definition 22.
LetS 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
(37)φ(x+y)=φ(x)+φ(y),φ(xγy)=φ(x)f(γ)φ(y),
for all x,y∈S and γ∈Γ, then we say (φ,f) is a homomorphismfrom S to S-. Also, if φ is a bijection, then (φ,f) is called an isomorphism, and S and S- are isomorphic.Definition 23.
LetA=(μA,λA) and B=(μB,λB) be two non-empty intuitionistic fuzzy  subsets of S and S_, respectively, and let φ:S→S- be a map. Then, (1)
theinverse image of B under φ is defined by
(38)φ-1(B)=(φ-1(μB),φ-1(λB)),
where
(39)φ-1(μB)=μB∘φ,φ-1(λB)=λB∘φ,(2)
theimage of A under φ, denoted by φ(A), is the intuitionistic fuzzy set in S- defined by φ(A)=(φ(μA),φ*(λA)), where for each y∈S-(40)φ(μA)(y)={supx∈φ-1(y)μA(x)ifφ-1(y)≠∅,0otherwise,φ*(λA)(y)={infx∈φ-1(y)λA(x)ifφ-1(y)≠∅,1otherwise.Now, the next theorem is about image and preimage of intuitionistic fuzzy sets.Theorem 24.
LetS be a Γ-semihyperring, let S- be a Γ--semihyperring, and let (φ,f) be a homomorphism from S to S-. Then,(1)
ifA=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, then φ(A) is an intuitionistic fuzzy Γ--hyperideal ofS-,(2)
ifB=(μB,λB) 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 NoetherianΓ-Semihypergroups

In this section, we give some characterizations of Artinian and NoetherianΓ-semihyperrings.Definition 25.
LetS 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
(41)I1⊆I2⊆I3⊆⋯⊆Ii⋯(I1⊇I2⊇I3⊇⋯⊇Ii⋯),
there exists n∈ℕ, such that Ii=Ii+1, for all i≥n.Proposition 26 (see [28]).
LetS be a Γ-semihypering. Then, the following conditions are equivalent:(1)
S is Noetherian.(2)
S satisfies the maximum condition for Γ-hyperideals.(3)
EveryΓ-hyperideal of S is finitely generated.Theorem 27.
If every intuitionistic fuzzyΓ-hyperideal of Γ-semihypering has finite number of values, then S is Artinian.Proof.
Suppose that every intuitionistic fuzzyΓ-hyperideal of Γ-semihyperings has finite number of values, and S is not Artinian. So, there exists a strictly descending chain
(42)S=I0⊇I1⊇I2⊇⋯
of Γ-hyperideals of S. We define the intuitionistic fuzzy set A by
(43)μA(x)={nn+1ifx∈In-In+1,1ifx∈⋂n=0∞In,λA(x)={1n+1ifx∈In-In+1,0ifx∈⋂n=0∞In.
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.
LetS be a Γ-semihypering. Then the following statements are equivalent:(1)
S is Noetherian.(2)
The set of values of any intuitionistic fuzzyΓ-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 {tn}, such that μA(x)=tn and λA(x)=1-tn. Let In={x∈S∣μ(x)≥tn} and Jn={x∈S∣λ(x)≤1-tn}. 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∈ℕIn. It is easy to see that I is a Γ-hyperideal of S.  Now, we define
(44)μA(x)={0ifx∉I,1kwherek=min{n∈ℕ∣x∈In},λA(x)={1ifx∉I,k-1k+1wherek=max{n∈ℕ∣x∈In}.
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 aΓ-semihyperring S both Artinian and Noetherian, then every intuitionistic fuzzy Γ-hyperideal of S is finite valued.Proof.
Suppose thatA is an intuitionistic fuzzy Γ-hyperideal of S, Im(μA) and Im(λA) 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(μA) and s1>s2>s3⋯ is strictly decreasing sequence in Im(λA). Now, we obtain
(45)U(μA;t1)⊃U(μA;t2)⊃U(μA;t3)⊃⋯,L(λA;s1)⊃L(λA;s2)⊃L(λA;s3)⊃⋯
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(μA;ti)=U(μA;ti+n) and L(λA;si)=L(λA;si+n) for n≥1. This means that ti=ti+n and si=si+n. This is a contradiction.
Case 2. Assume that t1>t2>t3>⋯, s1<s2<s3⋯ are strictly decreasing and increasing sequence in Im(μA) and Im(λA), respectively. From these equalities we conclude that
(46)U(μA;t1)⊂U(μA;t2)⊂U(μA;t3)⊂⋯,L(λA;s1)⊂L(λA;s2)⊂L(λA;s3)⊂⋯
are strictly ascending and descending chains of Γ-hyperideals, respectively. Because of the hypothesis there exists a natural number j, such that U(μA;ti)=U(μA;ti+n) and L(λA;si)=L(λA;si+n) for n≥1. This implies that ti=ti+n and si=si+n which is a contradiction.



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