# FuzzyΓ-Hyperideals in Γ-Hypersemirings by Using Triangular Norms

Authors: B. A. Ersoy; B. Davvaz; S. Onar; V. Leoreanu-Fotea
Journal: The Scientific World Journal
 (2014)

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## Abstract

The concept ofΓ-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.


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## Body

## 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 ΓN-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 μA:X→[0,1] which associates with each point x∈X its grade or degree of membership μμ(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 Hv-ideals of Hv-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

LetH 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∈B‍a∘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∘y‍u∘z=⋃v∈y∘z‍x∘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 1R∈R means that 1R·x=x·1R=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 at-norm and its dual operation is due to Schweizer and Sklar [27]. By at-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 everyt-norm T, we set ΔT={x∈[0,1],T(x,x)=x}. At-norm on [0,1] is called a continuoust-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 continuoust-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 andt-conorms differ only with respect to their respective boundary conditions. In fact, the concepts oft-norms andt-conorms are dual in some sense. Anthony and Sherwood [28] redefined a fuzzy subgroup of a group by using the notion oft-norm.


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

In this section, we define the notion ofT-fuzzy sub-Γ-semihyperrings and T-fuzzy Γ-hyperideals of a Γ-semihyperring and we study some of their properties. Let T be at-norm and μ be a fuzzy subset of a Γ-semihyperring R. Then, we say μ has imaginable property if Im⁡μ⊆ΔT.Definition 1.
LetR 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.AT-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 thatR=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
(1)μ(x)={34ifx∈Γ59otherwise
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
(2)inf⁡z∈x+y{μ(z)}=min⁡{μ(x),μ(y)}={34ifx,y∈Γ59otherwise,inf⁡z∈xγy{μ(z)}=min⁡{μ(x),μ(γ),μ(y)}=59.
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
(3)T(μ(x),μ(y))≤inf⁡z∈x+y{μ(z)},T(μ(x),μ(y))≤inf⁡z∈xγy{μ(z)}.
Therefore, μ is a T-fuzzy sub-Γ-semihyperring of R.Lemma 3.
LetR be a Γ-semihyperring, T be a t-norm, and μ be a T-fuzzy sub-Γ-semihyperring of S. Then
(4)Tn(μ(x1),…,μ(xn))≤inf⁡z∈x1+⋯+xn{μ(z)},Tn(μ(x1),…,μ(xn))≤inf⁡z∈x1γ⋯γxn{μ(z)},
for all x1,…,xn∈R and γ∈Γ, where
(5)Tn(t1,…,tn)={t1ifn=1T(t1,t2)ifn=2T(ti,Tn-1(t1,…,ti-1,ti+1,…,tn))ifn>2.Proof.
The proof is straightforward by mathematical induction.Lemma 4.
LetR 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
(6)T(inf⁡a∈A{μ(a)},inf⁡b∈B{μ(b)})≤inf⁡z∈A+B{μ(z)},T(inf⁡a∈A{μ(a)},inf⁡b∈B{μ(b)})≤inf⁡z∈AγB{μ(z)},
for all γ∈Γ.Proof.
The proof is straightforward.Theorem 5.
LetR 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∈ΔT and 0<a≤b.Proof.
(
1
)

⇒

(
2
): Suppose that a∈ΔT 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∈ΔT 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⊆ΔT, both μ(x) and μ(y) are in ΔT. Now, we have
(7)T(T(μ(x),μ(y)),T(μ(x),μ(y)))=T(T(μ(x),T(μ(y),μ(x))),μ(y))=T(T(μ(x),T(μ(x),μ(y))),μ(y))=T(T(μ(x),μ(x)),T(μ(y),μ(y)))=T(μ(x),μ(y)),
and so T(μ(x),μ(y))∈ΔT. Assume that a=T(μ(x),μ(y)). If a=0, then
(8)T(μ(x),μ(y))=0≤inf⁡z∈x+y{μ(z)},T(μ(x),μ(y))=0≤inf⁡z∈xγy{μ(z)}.
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.
LetR 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
(9)T(μ(x),μ(y))≤inf⁡z∈x+y{μ(z)},∀x,y∈R,μ(y)≤inf⁡z∈xγy{μ(z)},∀x,y∈R,∀γ∈Γ.(2)
μ is called a T-fuzzy right Γ-hyperideal of R if
(10)T(μ(x),μ(y))≤inf⁡z∈x+y{μ(z)},∀x,y∈R,μ(x)≤inf⁡z∈xγy{μ(z)},∀x,y∈R,∀γ∈Γ.(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.
LetR 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∈ΔT and 0<a≤b.Proof.
The proof is similar to the proof of Theorem5.Letμ 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.
LetR 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.
LetA be a subset of R. Then(1)
the characteristic functionχA is a T-fuzzy sub-Γ-semihyperring of R if and only if A is a sub-Γ-semihyperring of R;(2)
the characteristic functionχA is a T-fuzzy Γ-hyperideal of R if and only if A is a Γ-hyperideal of R.Theorem 10.
LetR 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
(11)μ(x)={rifx∈Ksotherwise,
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.
LetR1 and R2 be Γ1 and Γ2-semihyperrings, respectively. If there exists a map φ:R1→R2 and a bijection f:Γ1→Γ2 such that
(12)φ(x+y)={φ(z)∣z∈x+y}=φ(x)+φ(y),φ(xγy)={φ(z)∣z∈xγy}=φ(x)f(γ)φ(y),
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.
LetR1 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 thatx,y∈R1 and γ∈Γ. Then, we have
(13)inf⁡z∈x+y{φ-1(λ)(z)}=inf⁡z∈x+y{λ(φ(z))}≥inf⁡φ(z)∈φ(x+y){λ(φ(z))}≥inf⁡φ(z)∈φ(x)+φ(y){λ(φ(z))}≥T(λ(φ(x)),λ(φ(y)))=T(φ-1(λ)(x),φ-1(λ)(y));inf⁡z∈xγy{φ-1(λ)(z)}=inf⁡z∈xγy{λ(φ(z))}≥inf⁡φ(z)∈φ(xγy){λ(φ(z))}≥inf⁡φ(z)∈φ(x)f(γ)φ(y){λ(φ(z))}≥T(λ(φ(x)),λ(φ(y)))=T(φ-1(λ)(x),φ-1(λ)(y)).
Therefore, φ-1(λ) is a T-fuzzy sub-Γ-semihyperring of R1.Proposition 13.
LetR1 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 Proposition12.Let{ai}i∈I and {bj}j∈J be two sets of real numbers in [0,1]. Then, we say T is infinitely distributive if
(14)T(sup⁡i∈I{ai},sup⁡j∈J{bj})=sup⁡i∈Ij∈J⁡{T(ai,bj)}.
If T is continuous, then T is infinitely distributive [30].Lemma 14.
LetT be a continuous t-norm and {μi}i∈I be a family of T-fuzzy sub-Γ-semihyperring of R. Then, ⋂i∈I‍μi is a T-fuzzy sub-Γ-semihyperring of R.Proof.
For anyx,y∈R and γ∈Γ, we have
(15)inf⁡z∈x+y{(⋂‍i∈Iμi)(z)}=inf⁡z∈x+y{inf⁡i∈I{μi(z)}}=inf⁡i∈I{inf⁡z∈x+y{μi(z)}}≥inf⁡i∈I{T(μi(x),μi(y))}=T(inf⁡i∈I{μi(x)},inf⁡i∈I{μi(y)})=T((⋂‍i∈Iμi)(x),(⋂‍i∈Iμi)(y));inf⁡z∈xγy{(⋂‍i∈Iμi)(z)}=inf⁡z∈xγy{inf⁡i∈I{μi(z)}}=inf⁡i∈I{inf⁡z∈xγy{μi(z)}}≥inf⁡i∈I{T(μi(x),μi(y))}=T(inf⁡i∈I{μi(x)},inf⁡i∈I{μi(y)})=T((⋂‍i∈Iμi)(x),(⋂‍i∈Iμi)(y)).Lemma 15.
LetR1 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(μt-ϵ).Proof.
The proof is similar to the proof of Lemma  3.5 in [31].Proposition 16.
LetR1 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 thatμ 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.
LetR1 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.
LetR1 and R2 be two Γ-semihyperrings and let μ and λ be fuzzy subsets of R1 and R2, respectively. Then(1)
ifμandλareT-fuzzy sub-Γ-semihyperrings ofR1andR2, respectively, thenμ×λis aT-fuzzy sub-Γ-semihyperring ofR1×R2;(2)
ifμandλareT-fuzzyΓ-hyperideals ofR1andR2, respectively, thenμ×λis aT-fuzzyΓ-hyperideal ofR1×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]).
LetR be a Γ-semihyperring. Then, the following conditions are equivalent:(1)
Rsatisfying 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]).
AΓ-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]).
LetAn=[n,nn+1] for every n∈Z, R=⋃n∈Z‍An, and Γ=Z. Then, R is a Noetherian Γ-semihyperring with respect to the following hyperoperations:
(16)x⊕y=An+m,xγy=Anγm,
where x∈An and y∈Am.Theorem 22.
Let{Ak∣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
(17)μ(x)={kk+1ifx∈Ak∖Ak+1,k=0,1,2,…1ifx∈⋂k=0∞‍Ak,
for all x∈R, where A0 stands for R. Let T be at-norm with Im μ⊆ΔT. Then, μ is a T-fuzzy Γ-hyperideal of R.Proof.
Letx,y∈R. Suppose that x∈Ak∖Ak+1 and y∈Ar∖Ar+1 for k=0,1,2,… and r=0,1,2,…. Without loss of generality we may assume that k≤r. Then, obviously y∈Ak. Since Ak is a Γ-hyperideal of R, it follows that x+y⊆Ak and xαy⊆Ak which imply that inf⁡z∈x+yμ(z)≥k/(k+1)=T(μ(x),μ(y)) and inf⁡z∈xαyμ(z)≥k/(k+1)=μ(y) for all α∈Γ. If x∈⋂k=0∞‍Ak and y∈⋂k=0∞‍Ak, then x+y⊆⋂k=0∞‍Ak. Hence, inf⁡z∈x+yμ(z)=1=T(μ(x),μ(y)) and inf⁡z∈xαyμ(z)=1≥μ(y). If x∉⋂k=0∞‍Ak and y∈⋂k=0∞‍Ak, then there exists n∈N such that x∈An∖An+1. It follows that x+y⊆An which implies that inf⁡z∈x+yμ(z)≥n/(n+1)=T(μ(x),μ(y)) and inf⁡z∈xαyμ(z)≥n/(n+1)=μ(y) for all α∈Γ. Finally, assume that x∈⋂k=0∞‍Ak and y∉⋂k=0∞‍Ak, then y∈An∖An+1 for some n∈N. Therefore, x+y⊆An, and thus inf⁡z∈x+yμ(z)≥n/(n+1)=T(μ(x),μ(y)) and inf⁡z∈xαyμ(z)≥n/(n+1)=μ(y), for all α∈Γ. Hence, μ is a T-fuzzy Γ-hyperideal of R.Theorem 23.
LetR 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⊆ΔT. 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{tk} be a strictly increasing sequence of elements of Im⁡μ. Then 0≤t1≤t2≤⋯≤1. Then, U(μ;tr) is an ideal of M for all r=2,3,… Let x∈U(μ;tr). Then μ(x)≥tr≥tr-1, and so x∈U(μ;tr-1). Hence U(μ;tr)⊆U(μ;tr-1). Since tr-1∈Im⁡μ, there exists xr-1∈M such that μ(xr-1)=tr-1. It follows that xr-1∈U(μ;tr-1) but xr-1∉U(μ;tr). Thus, U(μ;tr)⊂U(μ;tr-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.
LetR be a Γ-semiring, μ be a fuzzy subset of R, and T be a t-norm with Im μ⊆ΔT. Then, the following conditions are equivalent: (1)
R is a Noetherian Γ-semihyperring,(2)
the set of values of anyΓ-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 {tk} such that μ(xk)=tk. It follows that U(μ;t1)⊂U(μ;t2)⊂⋯ is a strictly ascending chain of Γ-hyperideals of M, where U(μ;tr)={x∈M∣μ(x)≥tr}, 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
(18)A1⊂A2⊂A3⊂⋯
of Γ-hyperideals of R. Note that A=⋂k=0∞‍Ak is a Γ-hyperideal of R. Define a fuzzy subset in R by
(19)μ(x)={αifx∉A,1rwherer=min⁡{k∈N∣x∈Ak}.
We claim that μ is a T-fuzzy Γ-hyperideal of R. Let x,y∈R. If x∈Ak∖Ak-1 and y∈Ak∖Ak-1, then x-y⊂Ak and xαy⊂Ak. It follows that inf⁡z∈x+yμ(z)=1/k=T(μ(x),μ(y)) and inf⁡z∈xαyμ(z)≥1/k≥μ(y), for all β∈Γ. Suppose that x∈Ak and y∈Ak∖Ar for all r<k. Since Ak is a Γ-hyperideal of R, it follows that x+y⊂Ak. Hence, inf⁡z∈x+yμ(z)≥1/k≥1/(k+1)≥μ(y) and so inf⁡z∈x+yμ(z)≥min⁡{μ(x),μ(y)}≥T(μ(x),μ(y)). Similarly, for the case x∈Ak∖Ar and y∈Ak, we have inf⁡z∈x+yμ(z)≥min⁡{μ(x),μ(y)}≥T(μ(x),μ(y)) and inf⁡z∈xαyμ(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{μα∣α∈Λ} of fuzzy subsets in R, we define the join ∨α∈Λμα and the meet ∨α∈Λμα as follows:
(20)(⋁α∈Λμα)(x)=sup⁡⁡{μα(x)∣α∈Λ},(⋀α∈Λμα)(x)=inf⁡⁡{μα(x)∣α∈Λ},
for all x∈R, where Λ is any index set.Theorem 25.
The family ofT-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∈Λμα and ∧α∈Λμα are T-fuzzy Γ-hyperideals of R for family {μα∣α∈Λ} of T-fuzzy Γ-hyperideals of R. For any x,y∈R, we have
(21)inf⁡z∈x+y(⋁α∈Λμα)(z)≥sup⁡{T(μα(x),μα(y))∣α∈Λ}≥T(sup⁡⁡{μα(x)∣α∈Λ},kkkkksup⁡⁡{μα(y)∣α∈Λ})=T((⋁α∈Λμα)(x),(⋁α∈Λμα)(y)),inf⁡z∈x+y(⋀α∈Λμα)(z)≥inf⁡{T(μα(x),μα(y))∣α∈Λ}=T((⋀α∈Λμα)(x),(⋀α∈Λμα)(y)).
Now, letx,y∈M and β∈Γ. Then
(22)inf⁡z∈xβy(⋁α∈Λμα)(z)≥sup⁡{μα(y)∣α∈Λ}=(⋁α∈Λμα)(y),(23)inf⁡z∈xβy(⋀α∈Λμα)(z)≥inf⁡{μα(y)∣α∈Λ}=(⋀α∈Λμα)(y).
Hence, ∨α∈Λμα and ∧α∈Λμα are T-fuzzy Γ-hyperideals of R. This completes the proof.



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*Source: 428635-2014-05-20.xml*