AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 560698 10.1155/2013/560698 560698 Research Article Structure of Intuitionistic Fuzzy Sets in Γ-Semihyperrings Ersoy Bayram Ali 1 Davvaz Bijan 2 Başar Feyzi 1 Department of Mathematics Yildiz Technical University 34210 Istanbul Turkey yildiz.edu.tr 2 Department of Mathematics Yazd University P.O. Box 89195-741, Yazd Iran yazduni.ac.ir 2013 2 2 2013 2013 21 10 2012 27 12 2012 2013 Copyright © 2013 Bayram Ali Ersoy and Bijan Davvaz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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  has achieved a great success in various fields. In 1971, Rosenfeld  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 . The concept of intuitionistic fuzzy set was introduced and studied by Atanassov  as a generalization of the notion of fuzzy set. In , Biswas studied the notion of an intuitionistic fuzzy subgroup of a group. In , Kim and Jun introduced the concept of intuitionistic fuzzy ideals of semirings. Also, in , 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  weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes , Luh , and Kyuno  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 .

Algebraic hyperstructures represent a natural extension of classical algebraic structures, and they were introduced by the French mathematician Marty . 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 .

In , 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 , 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 . In , Davvaz introduced the notion of fuzzy subhypergroups as a generalization of fuzzy subgroups, and this topic was continued by himself and others. In , Leoreanu-Fotea and Davvaz studied fuzzy hyperrings. Recently, Davvaz et al.  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 (1)AB=aA,bBab,  xA={x}A,Ax=A{x}.

A hypergrupoid (S,) is called a semihypergroup if for all x,y,andz of S we have (xy)z=x(yz). That is, (2)uxyuz=vyzxv. A semihypergroup (S,) is called a hypergroup if for all xS, xS=Sx=S.

A semihyperring is an algebraic structure (S,+,·) which satisfies the following properties:

(S,+) is a commutative semihypergroup; that is,

(x+y)+z=x+(y+z),

x+y=y+x,  for all x,y,zS.

(S,·) is a semihypergroup; that is, (x·y)·z=x·(y·z), for all x,y,zS.

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,zS.

The element 0S is an absorbing element; that is, 0·x=x·0=0, for all xS. A semihyperring (S,+,·) is called commutative if x·y=y·x for all x,yS. Vougiouklis in  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 1SS, such that 1S·x=x·1S=x, for all xS. An element xS is called unit if there exists yS, such that y·x=x·y=1S. A non-empty subset A of a semihyperring (S,+,·) is called subsemihyperring if x+yA and x·yA, for all x,yA. A left hyperideal of a semihyperring S is a non-empty subset I of S satisfying the following:

x+yI, for all x,yI,

x·aI, for all aI and xS.

Let S and Γ be two non-empty sets. Then, S is called a Γ-semihypergroup; if for every hyperoperation γΓ, α,βΓ, and x,y,zS, we have (4)(xβy)γz=xβ(yγz).

For example, S=[0,1] and ΓS. For all αΓ and for all x,yS, 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 . We recall the following definition from .

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:

xα(y+z)=xαy+xαz,

(x+y)αz=xαz+yαz,

x(α+β)z=xαz+xβz,

xα(yβz)=(xαy)βz,

for all x,y,zS and α,βΓ.

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,yS and αΓ. We say that a Γ-semihyperring S is with zero, if there exists 0S, such that xx+0 and 00αx, for all xS and αΓ. Let A and B be two non-empty subsets of a Γ-semihyperring and xS. We define (5)A+B={xxa+b,aA,bB},AΓB={xxaαb,aA,bB,αΓ}.

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 (6)S1+S1S1,S1ΓS1S1.

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,yS and γΓ. Then, it is not difficult to see that S is a multiplicative Γ-semihyperring.

Example 3.

Let S=+, Γ={γii} and Ai=i+. We define xγiyxAiy for every γΓ and x,yS. 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)ABC={ZMm,n(S)ZABC}, for all A,CMm,n(S) and BMn,m(S).

Then, Mm,n(S) is Mn,m(S)-semihyperring . Now, suppose that (9)X={(aij)m,na110,a210,aij=0otherwise}. 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,I2I implies I1+I2I and IΓSI    (SΓII) and is a Γ-hyperideal of S if it is both left and right Γ-hyperideal.

Example 5.

Consider the following: (10)S={[abcd]a,b,c,d+{0}},Ai={[ie00if]e,f2+},i,Γ={γii}. Then, S is a Γ-semihyperring under the matrix addition and the hyperoperation as follows: (11)xγiy=xAiy, for all x,yS and γiΓ. Let (12)I1={[ab00]a,b+{0}},I2={[a0c0]a,c+{0}},I3={[abcd]a,b,c,d2+{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 subset μ of a non-empty set S is a function from S to [0,1]. For all xS  μc, 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 μλ and μλ, are defined by (13)(μλ)(x)=min{μ(x),λ(x)},(μλ)(x)=max{μ(x),λ(x)}.

Definition 6.

Let  S be a Γ-semihyperring, and let μ be a fuzzy subset of S. Then,

μ is called a fuzzy left Γ-hyperideal of S if (14)min{μ(x),μ(y)}infzx+y{μ(z)},μ(y)infzxγy{μ(z)}x,yS,γΓ.

μ is called a fuzzy right Γ-hyperideal of S if (15)min{μ(x),μ(y)}infzx+y{μ(z)},μ(x)infzxγy{μ(z)}x,yS,γΓ.

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=6, Γ={γ2,γ3}, S2={0-,3-}, and S3={0-} be non-empty subsets of S1. We define xγiy=xSiy, for every γiΓ and x,yS1. 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 Fuzzy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M236"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math></inline-formula>-Hyperideals

The concept of intuitionistic fuzzy set was introduced and studied by Atanassov . 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))xS}. Here, μA:  S[0,1] is the degree of membership of the element xS to the set A, and λA:  S[0,1] is the degree of nonmembership of the element xS to the set A. We have also 0μA(x)+λA(x)1, for all xS.

Example 8 (see [<xref ref-type="bibr" rid="B15">35</xref>]).

Consider the universe X={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 by B, as follows: (18)μB(x)=(μA(x))2,λB(x)=1-(1-λA(x))2, for all xX. 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))xS}. Let A=(μA,λA) and B=(μB,λB) be two intuitionistic fuzzy sets of S. Then, the following expressions are defined in  as follows:

AB if and only if μA(x)μB(x) and λA(x)λB(x), for all xS,

AC={(x,λA(x),μA(x))xS},

AB={(x,min{μA(x),μB(x)},max{λA(x),λB(x)})xS},

AB={(x,max{μA(x),μB(x)},min{λA(x),λB(x)})xS},

A={(x,μA(x),μAc(x))xS},

A={(x,λAc(x),λA(x))xS}.

Now, we introduce the notion of intuitionistic fuzzy Γ-hyperideals of Γ-semihyperrings.

Definition 9.

Let S be a Γ-semihyperring.

An intuitionistic fuzzy set A=(μA,λA) in S is called a left intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring S if

min{μA(x),μA(y)}infzx+y{μA(z)}  and max{λA(x),λA(y)}supzx+y{λA(z)},

μA(y)infzxγy{μA(z)} and supzxγy{λA(z)}λA(y),

for all x,yS,  γΓ.

An intuitionistic fuzzy set A=(μA,λA) in S is called a right intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring S if

min{μA(x),μA(y)}infzx+y{μA(z)}  and max{λA(x),λA(y)}supzx+y{λA(z)},

μA(x)infzxγy{μA(z)} and supzxγy{λA(z)}λA(x),

for all x,yS,  γΓ.

An intuitionistic fuzzy set A=(μA,λA) 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 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.

Let S be a Γ-semihyperring defined in Example 5. Suppose that (21)μAi(x)={58ifa,b,c,d2+{0},25otherwise,λAi(x)={19ifa,b,c,d2+{0},47otherwise, for all xS. Then, Ai=(μAi,λAi) is an intuitionistic fuzzy Γ-hyperideal of S.

Theorem 12.

Let A=(μA,λA) be an intuitionistic fuzzy Γ-hyperideal of Γ-semihyperring S and 0t1. 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 xS. Then, B=(μB,λB) is an intuitionistic fuzzy Γ-hyperideal of S.

Proof.

Note that 0μB(x)+λB(x)=tμA(x)+(1-t)λA(x)t+1-t=1.

Theorem 13.

Let A=(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 xS. Then, B=(μB,λB) is an intuitionistic fuzzy Γ-hyperideals of S.

Proof.

Since A=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal, we have infzx+y{μA(z)}min{μA(x),μA(y)} and so (22)(infzx+y{μA(z)})2(min{μA(x),μA(y)})2. Hence, infzx+y{(μA(z))2}min{(μA(x))2,(μA(y))2}; that is, (23)infzx+y{μB(z)}min{μB(x),μB(y)}. Since infzxγy{μA(z)}μA(y), so (infzxγy{μA(z)})2(μA(y))2 which implies that infzxγy{(μA(z))2}(μA(y))2. Hence, infzxγy{μB(z)}μB(y). Also, we have (24)supzx+y{λA(z)}max{λA(x),λA(y)}infzx+y{-λA(z)}min{-λA(x),-λA(y)}1-infzx+y{-λA(z)}1+min{-λA(x),-λA(y)}infzx+y{1-λA(z)}min{1-λA(x),1-λA(y)}(infzx+y{1-λA(z)})2(min{1-λA(x),1-λA(y)})2infzx+y{(1-λA(z))2}min{(1-λA(x))2,(1-λA(y))2}supzx+y{-(1-λA(z))2}max{-(1-λA(x))2,-(1-λA(y))2}1+supzx+y{-(1-λA(z))2}1+max{-(1-λA(x))2,-(1-λA(y))2}supzx+y{1-(1-λA(z))2}max{1-(1-λA(x))2,1-(1-λA(y))2}supzx+y{λB(z)}max{λB(x),λB(y)}. Moreover, we have (25)supzxγy{λA(z)}λA(y)infzxγy{-λA(z)}-λA(y)1-infzxγy{-λA(z)}1-λA(y)infzxγy{1-λA(z)}1-λA(y)(infzxγy{1-λA(z)})2(1-λA(y))2infzxγy{(1-λA(z))2}(1-λA(y))2supzx+y{-(1-λA(z))2}-(1-λA(y))21+supzx+y{-(1-λA(z))2}1-(1-λA(y))2supzx+y{1-(1-λA(z))2}1-(1-λA(y))2supzx+y{λB(z)}λB(y).

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 {Ai=(μAi,λAi)} is a collection of intuitionistic fuzzy Γ-hyperideals of S, then iΛAi and iΛAi are intuitionistic fuzzy Γ-hyperideals of S, too.

Proof.

The proof is straightforward.

Theorem 15.

An intuitionistic fuzzy subset A=(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 that A=(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)infzx+y{λAc(z)}=infzx+y{1-λA(z)}=1-supzx+y{λA(z)}min{1-λA(x),1-λA(y)}=min{λAc(x),λAc(y)},infzxγy{λAc(z)}=infzxγy{1-λA(z)}=1-supzxγy{λA(z)}1-λA(y)=λAc(y), for all x,yS and γΓ.

Conversely, suppose that μA and λAc are left (res. right) fuzzy Γ-hyperideals of S. We have min{μA(x),μA(y)}infzx+y{μA(z)} and μA(y)infzxγy{μA(z)}, for all x,yS and γΓ. We obtain (27)supzx+y{λA(z)}=supzx+y{1-λAc(z)}=1-infzx+y{λAc(z)}max{1-λAc(x),1-λAc(y)}=max{λA(x),λA(y)},supzxγy{λA(z)}=supzxγy{1-λAc(z)}=1-infzxγy{λAc(z)}1-λAc(y)=λA(y). Therefore, A=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S.

For any fuzzy set μ of S and any t[0,1], U(μ;t)={xSμ(x)t} is called an upper bound t-level    cut  of μ, and L(μ;t)={xSμ(x)t} is called a lower bound t-level    cut  of 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(μA;t) and L(λA;s) of S are left (res. right) Γ-hyperideals, when they are non-empty.

Proof.

Assume that A=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S. Let x,yU(μA;t). Since infzx+yμA(z)min{μA(x),μA(y)}t, then we have x+yU(μA;t). Let xS and yU(μA;t). We have infzxγyμA(z)μA(y)t. Therefore, zU(μA;t) and so xγyU(μA;t).

Now, let x,yL(λA;s). Since λA(x)s and λA(y)s, we have supzx+y{λA(z)}max{λA(x),λA(y)}s. Then, zL(λA;s), and so x+yL(λA;s). Now, for all xS, yL(λA;s) and γΓ we have supzxγy{λA(z)}λA(y)s. Therefore, zL(λA;s) and so xγyL(λA;s).

Conversely, suppose that all non-empty level sets  U(μA;t) and L(λA;s) are left Γ-hyperideals. Let x,yS and γΓ. Let t0=μA(x), t1=μA(y) and s0=λA(x), s1=λA(y) with t0t1, s0s1, then x,yU(μA;t0) and x,yL(λA;s1). Since (28)x+yU(μA;t0),x+yL(λA;s1), we have (29)t0=min{μA(x),μA(y)}infzx+y{μA(z)},s1=max{λA(x),λA(y)}supzx+y{λA(z)}. Since U(μA;t) and L(λA;t) are left Γ-hyperideals, and we have zxγyU(μA;t) and zxγyU(λA;s), we have μA(z)t0 and λA(z)s1. And so infzxγyμA(z)μA(y) and supzxγyλA(z)λA(y). Hence, A=(μA,λA) 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 μA and λA as follows: (30)μA={a0ifxI,a1ifxS-I,λA={b0ifxI,b1ifxS-I, where 0a1<a0,  0b0<b1 and ai+bi1 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.

If A=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, then so are A and A.

Proof.

Since A=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, we have (31)min{μA(x),μA(y)}infzx+y{μA(z)},μA(x)infzxγy{μA(z)}, for all x,y,zS and γΓ. Moreover, (32)supzx+y{μAc(z)}=supzx+y{1-μA(z)}=1-infzx+yμA(z)max{1-μA(x),1-μA(y)}=max{μAc(x),μAc(y)},supzxγy{μAc(z)}=supzxγy{1-μA(z)}=1-infzxγyμA(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 fuzzy Γ-hyperideal A, then A=A=A. While for a proper intuitionistic fuzzy Γ-hyperideal A, we have AAA and AAA.

Theorem 19.

If A=(μA,λA) is a left intuitionistic fuzzy Γ-hyperideal of S, then we have, (33)μA(x)=sup{t[0,1]xU(μA;t)},λA(x)=inf{s[0,1]xL(λA;s)}, for all xS.

Proof.

The proof is straightforward.

Let A=(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={supzxγy{min(θ1(x),σ1(y))x,yS,γΓ},0otherwise,t2={infzxγy{max(θ2(x),σ2(y))x,yS,γΓ},0otherwise.

Let χI 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.

If S is a Γ-semihyperring and A is an intuitionistic fuzzy Γ-hyperideal of S, then XΓAA, where X=(χS,χSc) and χS 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,

if w is a fixed element of S, then the set μAw={xSμ(x)μ(w)} and λAw={xSλ(x)λ(w)} are Γ-hyperideals of S,

the sets U={xSμA(x)=0} and L={xSλA(x)=0} are Γ-hyperideals of S.

Proof.

(1) Suppose that x,yμAw. Then, we have (36)infzx+yμA(z)min{μA(x),μA(y)}μ(w), which implies that zU(μA;μ(w)) and so x+yU(μA;μ(w)). Similarly, suppose that x,yλAw. We have supzx+yλA(z)max{λA(x),λA(y)}μ(w) which implies that zU(λA;λ(w)) and so x+yU(λA;λ(w)). Assume that xS, yμAw and γΓ. Then, we have infzxγyμA(z)μA(y)μ(w) which implies that zU(μA;μ(w)) and so xγyU(μA;μ(w)). Similarly, suppose that xλAw, yS and γΓ. We have supzxγyλA(z)λA(x)μ(w) which implies that zU(λA;λ(w)) and so xγyU(λA;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:SS- and a bijection f:ΓΓ-, such that (37)φ(x+y)=φ(x)+φ(y),φ(xγy)=φ(x)f(γ)φ(y), for all x,yS and γΓ, 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=(μA,λA) and B=(μB,λB) be two non-empty intuitionistic fuzzy subsets of S and S_, respectively, and let φ:SS- be a map. Then,

the inverse image of B under φ is defined by (38)φ-1(B)=(φ-1(μB),φ-1(λB)), where (39)φ-1(μB)=μBφ,φ-1(λB)=λBφ,

the image 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 yS-(40)φ(μA)(y)={supxφ-1(y)μA(x)ifφ-1(y),0otherwise,φ*(λA)(y)={infxφ-1(y)λA(x)ifφ-1(y),1otherwise.

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,

if A=(μA,λA) is an intuitionistic fuzzy Γ-hyperideal of S, then φ(A) is an intuitionistic fuzzy Γ--hyperideal ofS-,

if B=(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 Noetherian <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M633"><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi></mml:mrow></mml:math></inline-formula>-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 (41)I1I2I3Ii(I1I2I3Ii), there exists n, such that Ii=Ii+1, for all in.

Proposition 26 (see [<xref ref-type="bibr" rid="B18">28</xref>]).

Let S be a Γ-semihypering. Then, the following conditions are equivalent:

S is Noetherian.

S satisfies the maximum condition for Γ-hyperideals.

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 (42)S=I0I1I2 of Γ-hyperideals of S. We define the intuitionistic fuzzy set A by (43)μA(x)={nn+1ifxIn-In+1,1ifxn=0In,λA(x)={1n+1ifxIn-In+1,0ifxn=0In. 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 I0I1I2 infinitely descending chain of Γ-hyperideals of S.

Theorem 28.

Let S be a Γ-semihypering. Then the following statements are equivalent:

S is Noetherian.

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 {tn}, such that μA(x)=tn and λA(x)=1-tn. Let In={xSμ(x)tn} and Jn={xSλ(x)1-tn}. Then, I1I2 and J1J2 are strictly infinite decreasing chains of Γ-hyperideals of S. This contradicts with our hypothesis.

( 2 ) ( 1 ) Suppose that I1I2 is a strictly infinite ascending chain Γ-hyperideals of S. Let I=nIn. It is easy to see that I is a Γ-hyperideal of S. Now, we define (44)μA(x)={0ifxI,1kwhere  k=min{nxIn},λA(x)={1ifxI,k-1k+1where  k=max{nxIn}. 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(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 n1. 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 n1. This implies that ti=ti+n and si=si+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.

Zadeh L. A. Fuzzy sets Information and Computation 1965 8 338 353 MR0219427 ZBL0139.24606 Rosenfeld A. Fuzzy groups Journal of Mathematical Analysis and Applications 1971 35 512 517 MR0280636 10.1016/0022-247X(71)90199-5 ZBL0194.05501 Liu W. J. Fuzzy invariant subgroups and fuzzy ideals Fuzzy Sets and Systems 1982 8 2 133 139 10.1016/0165-0114(82)90003-3 MR666626 ZBL0488.20005 Atanassov K. T. Intuitionistic fuzzy sets Fuzzy Sets and Systems 1986 20 1 87 96 10.1016/S0165-0114(86)80034-3 MR852871 ZBL0631.03040 Atanassov K. T. New operations defined over the intuitionistic fuzzy sets Fuzzy Sets and Systems 1994 61 2 137 142 10.1016/0165-0114(94)90229-1 MR1262464 ZBL0824.04004 Atanassov K. T. Intuitionistic Fuzzy Sets, Theory and Applications 1999 35 Heidelberg, Germany Physica Studies in Fuzziness and Soft Computing MR1718470 Biswas R. Intuitionistic fuzzy subgroups Notes on Intuitionistic Fuzzy Sets 1997 3 2 3 11 MR1474880 ZBL1164.20373 Kim K. H. Jun Y. B. Intuitionistic fuzzy ideals of semigroups Indian Journal of Pure and Applied Mathematics 2002 33 4 443 449 MR1902685 ZBL1002.20048 Gunduz C. Davvaz B. The universal coefficient theorem in the category of intuitionistic fuzzy modules Utilitas Mathematica 2010 81 131 156 MR2654736 ZBL1222.18010 Kim K. H. On intuitionistic fuzzy subhypernear-rings of hypernear-rings Turkish Journal of Mathematics 2003 27 3 447 459 MR1995603 ZBL1046.08004 Uçkun M. Öztürk M. A. Jun Y. B. Intuitionistic fuzzy sets in Γ-semigroups Bulletin of the Korean Mathematical Society 2007 44 2 359 367 10.4134/BKMS.2007.44.2.359 MR2325037 ZBL1132.20047 Barnes W. E. On the Γ-rings of Nobusawa Pacific Journal of Mathematics 1966 18 411 422 MR0197517 10.2140/pjm.1966.18.411 Luh J. On the theory of simple Γ-rings The Michigan Mathematical Journal 1969 16 65 75 MR0241475 10.1307/mmj/1029000167 Kyuno S. On prime Γ-rings Pacific Journal of Mathematics 1978 75 1 185 190 MR0491845 10.2140/pjm.1978.75.185 ZBL0381.16022 Saha N. K. On Γ-semigroup. II Bulletin of the Calcutta Mathematical Society 1987 79 6 331 335 MR943215 Sen M. K. Saha N. K. On Γ-semigroup. I Bulletin of the Calcutta Mathematical Society 1986 78 3 180 186 MR851844 Rao M. M. K. Γ-semirings. I Southeast Asian Bulletin of Mathematics 1995 19 1 49 54 MR1323412 ZBL0846.16034 Marty F. Sur une generalization de la notion de groupe Proceedings of the 8th Scandinavian Congress of Mathematicians 1934 Stockholm, Sweden 45 49 Corsini P. Prolegomena of Hypergroup Theory 1993 Tricesimo, Italy Aviani Editore MR1237639 Corsini P. Leoreanu V. Applications of Hyperstructure Theory 2003 5 Dordrecht, The Netherlands Kluwer Academic Advances in Mathematics MR1980853 Davvaz B. . Leoreanu-Fotea V. Hyperring Theory and Applications 2007 115 Palm Harber, Fla, USA International Academic Press Vougiouklis T. Hyperstructures and Their Representations 1994 115 Palm Harbor, Fla, USA Hadronic Press MR1270451 Anvariyeh S. M. Mirvakili S. Davvaz B. On Γ-hyperideals in Γ-semihypergroups Carpathian Journal of Mathematics 2010 26 1 11 23 MR2674473 Anvariyeh S. M. Mirvakili S. Davvaz B. Pawlak's approximations in Γ-semihypergroups Computers & Mathematics with Applications 2010 60 1 45 53 10.1016/j.camwa.2010.04.028 MR2651882 Heidari D. Dehkordi S. O. Davvaz B. Γ-semihypergroups and their properties Scientific Bulletin A 2010 72 1 195 208 MR2675953 ZBL1197.20062 Dehkordi S. O. Davvaz B. Γ-semihyperrings: approximations and rough ideals Bulletin of the Malaysian Mathematical Sciences Society. In press Dehkordi S. O. Davvaz B. Γ-semihyperrings: ideals, homomorphisms and regular relations submitted Dehkordi S. O. Davvaz B. Ideal theory in Γ-semihyperrings submitted Hedayati H. Davvaz B. Fundamental relation on Γ-hyperrings Ars Combinatoria 2011 100 381 394 MR2829483 Davvaz B. Fuzzy Hν-groups Fuzzy Sets and Systems 1999 101 1 191 195 10.1016/S0165-0114(97)00071-7 MR1658991 Leoreanu-Fotea V. Davvaz B. Fuzzy hyperrings Fuzzy Sets and Systems 2009 160 16 2366 2378 10.1016/j.fss.2008.11.007 MR2554457 ZBL1179.16029 Cristea I. Davvaz B. Atanassov's intuitionistic fuzzy grade of hypergroups Information Sciences 2010 180 8 1506 1517 10.1016/j.ins.2010.01.002 MR2587920 ZBL1192.20065 Davvaz B. Leoreanu-Fotea V. Intuitionistic fuzzy n-ary hypergroups Journal of Multiple-Valued Logic and Soft Computing 2010 16 1-2 87 103 MR2662788 Davvaz B. Corsini P. Leoreanu-Fotea V. Atanassov's intuitionistic (S, T)-fuzzy n-ary sub-hypergroups and their properties Information Sciences 2009 179 5 654 666 10.1016/j.ins.2008.10.023 MR2490199 Davvaz B. Majumder S. K. Atanassov's intuitionistic fuzzy interior ideals of Γ-semigroups Scientific Bulletin A 2011 73 3 45 60 MR2871879