The Scientific World Journal

Volume 2014 (2014), Article ID 428635, 8 pages

http://dx.doi.org/10.1155/2014/428635

Research Article

Fuzzy -Hyperideals in -Hypersemirings by Using Triangular Norms

B. A. Ersoy,¹ B. Davvaz,² S. Onar,¹ and V. Leoreanu-Fotea³

¹Department of Mathematics, Yildiz Technical University, 81270 Istanbul, Turkey

²Department of Mathematics, Yazd University, Yazd, Iran

³Faculty of Mathematics, “Al.I. Cuza” University, Iasi, Romania

Received 26 February 2014; Accepted 3 May 2014; Published 20 May 2014

Academic Editor: Anca Croitoru

Copyright © 2014 B. A. Ersoy et al. 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.

Abstract.

The concept of -semihyperrings was introduced by Dehkordi and Davvaz as a generalization of semirings, semihyperrings, and -semiring. In this paper, by using the notion of triangular norms, we define the concept of triangular fuzzy sub--semihyperrings as well as triangular fuzzy -hyperideals of a -semihyperring, and we study a few results in this respect.

1. Introduction

In [1], Nobusawa introduced -rings as a generalization of ternary rings. Let be an additive group whose elements are denoted by and another additive group whose elements are . Suppose that is defined to be an element of and that is defined to be an element of for every , and . If the products satisfy the following three conditions: , , ; ; (3) if for any and in , then ; then is called a -ring in the sense of Nobusawa [1]. Barnes [2] weakened slightly the conditions in the definition of -ring and gave a new definition of a -ring. Let and be two additive abelian groups. Suppose that there is a mapping from (sending into such that   , , ;   ; then is called a -ring in the sense of Barnes [2]. Nowadays, -rings mean the -rings due to Barnes and other -rings are known as -rings, that is, gamma rings in the sense of Nobusawa. Barnes [2], Luh [3], and Kyuno [4] studied the structure of -rings and obtained various generalization analogous to corresponding parts in ring theory. The notion of -semirings was introduced by Rao [5] as a generalization of semirings as well as -rings. Subsequently, by introducing the notion of operator semirings of a -semiring, Dutta and Sardar [6] enriched the theory of -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 -semihyperring as a generalization of semiring, semihyperring, and -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 be a set. A fuzzy subset of is characterized by a membership function which associates with each point its grade or degree of membership . 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 or . 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 -rings and studied a few properties. In [6], Dutta and Sardar studied the structures of fuzzy ideals of -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 -ideals of -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 -hyperideals of -semihyperring. Now, in this paper, we define the concept of triangular fuzzy sub--semihyperrings and fuzzy -hyperideals of a -semihyperring by using triangular norms, and we study a few results in this respect.

2. Basic Concepts

Let be a nonempty set and let be the set of all nonempty subsets of . A hyperoperation on is a map and the couple is called a hypergroupoid. If and are nonempty subsets of , then we denote , and . A hypergroupoid is called a semihypergroup if for all of we have , which means that . A semihyperring is an algebraic structure which satisfies the following properties:    is a commutative semihypergroup; that is, and for all ;    is a semihypergroup; the hyperoperation is distributive with respect to the hyperoperation +; that is, for all ; the element is an absorbing element; that is for all . A semihyperring is called commutative if and only if for all . 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 with identity means that for all . The concept of -semihyperring is introduced and studied by Dehkordi and Davvaz [11]. Let be a commutative semihypergroup and be a commutative group. Then, is called a -semihyperring if there exists a map (image to be denoted by for and ) and is the set of all nonempty subsets of satisfying the following conditions:

(1), (2), (3), (4),

for all and for all . One can find many examples of -semihyperrings in [11, 12]. In the above definition if is a semigroup, then is called a multiplicative -semihyperring. A -semihyperring is called commutative if for every and . We say that -semihyperring with zero if there exists such that , for all and . Let and be two nonempty subsets of -semihyperring and . We define and . A nonempty subset of -semihyperring is called a sub--semihyperring if it is closed with respect to the multiplication and addition. In other words, a nonempty subset of -semihyperring is a sub--semihypergroup if and .

A right (left) -hyperideal of a -semihyperring is an additive sub-semihypergroup such that (). If is both right and left -hyperideal of , then we say that is a two-sided -hyperideal or simply a -hyperideal of . In [22, 23], Ersoy and Davvaz studied fuzzy -hyperideals of -semihyperrings. We recall the notion of a fuzzy -hyperideal of a -semihyperring. Let be a -semihyperring and be a fuzzy subset of . Then    is called a fuzzy left -hyperideal of if and for all and ;    is called a fuzzy right -hyperideal of if and for all and ; (3)   is called a fuzzy -hyperideal of if it is both a fuzzy left -hyperideal and a fuzzy right -hyperideal of .

The concept of a triangular norm was introduced by Menger [26] in order to generalize the triangular inequality of a metric. The current notion of a t-norm and its dual operation is due to Schweizer and Sklar [27]. By a t-norm we mean a function satisfying the following conditions: (T1) ; (T2) ; (T3) if ; (T4) , for all . For every t-norm , we set . A t-norm on is called a continuous t-norm if is a continuous function from to with respect to the usual topology. Note that the function “Min” is a continuous t-norm. A triangular conorm (t-norm for short) is a binary operation on the unit interval , that is, a function , which for all satisfies (T1)–(T3) and (S4) . From an axiomatic point of view, t-norms and t-conorms differ only with respect to their respective boundary conditions. In fact, the concepts of t-norms and t-conorms are dual in some sense. Anthony and Sherwood [28] redefined a fuzzy subgroup of a group by using the notion of t-norm.

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

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

Definition 1. Let be a -semihyperring, be a -norm, and be a fuzzy subset of . Then, is called a -fuzzy sub--semihyperring of if

(1),(2),

for all and for all .

A -fuzzy sub--semihyperring of is said to be imaginable if it satisfies the imaginable property. Clearly, if is a -semiring, then is a -fuzzy sub--semiring of when

 (1′), (2′),

for all and for all .

Example 2. Suppose that , the set of natural numbers, and is a nonempty subset of . For any and , we define and . Then, is a -semihyperring. We define the fuzzy subset of by

and we consider the -norm , where . Then, for any and , we have

On the other hand, we have the following cases:

(1), (2) and ,(3).

Regarding the above cases, we have , , and . Thus, in every case, we obtain

Therefore, is a -fuzzy sub--semihyperring of .

Lemma 3. Let be a -semihyperring, be a -norm, and be a -fuzzy sub--semihyperring of . Then

for all and , where

Proof. The proof is straightforward by mathematical induction.

Lemma 4. Let be a -semihyperring, be a -norm, and be a -fuzzy sub--semihyperring of . Let and be nonempty subsets of . Then

for all .

Proof. The proof is straightforward.

Theorem 5. Let be a -semihyperring, be a -norm, and be a fuzzy subset of with imaginable property and the maximum of  . Then, the following two statements are equivalent:

(1) is a -fuzzy sub--semihyperring of ,(2) is a sub--semihyperring of whenever and .

Proof. : Suppose that and . If , then , which implies that . Similarly, assume that and . If and , then . Then, we have , and so is a sub--semihyperring of .

: Suppose that and . Since , both and are in . Now, we have

and so . Assume that . If , then

Now, let . Hence , which implies that and . Therefore and .

Definition 6. Let be a -semihyperring, be a -norm, and be a fuzzy subset of . Then

(1) is called a -fuzzy left -hyperideal of if

(2) is called a -fuzzy right -hyperideal of if

(3) is called a -fuzzy -hyperideal of if it is both a -fuzzy left -hyperideal and a -fuzzy right -hyperideal of .

Theorem 7. Let be a -semihyperring, be a -norm, and be a fuzzy subset of with imaginable property and the maximum of  . Then, the following two statements are equivalent:

(1) is a -fuzzy -hyperideal of ,(2) is a -hyperideal of whenever and .

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

Let be a fuzzy subset of and . The set is called a level subset of . So, we obtain the following corollary.

Corollary 8. Let be a -semihyperring and be a fuzzy subset of . Then

(1) is a Min-fuzzy sub--semihyperring of if and only if every nonempty level subset is a sub--semihyperring of ;(2) is a Min-fuzzy -hyperideal of if and only if every nonempty level subset is a -hyperideal of .

Corollary 9. Let be a subset of . Then

(1)the characteristic function is a -fuzzy sub--semihyperring of if and only if is a sub--semihyperring of ;(2)the characteristic function is a -fuzzy -hyperideal of if and only if is a -hyperideal of .

Theorem 10. Let be a -semihyperring and be a sub--semihyperring of . Let be the -norm defined by and be a fuzzy subset of defined by

for all and , where such that . Then, is a -fuzzy sub--semihyperring of . In particular, if and , then is imaginable.

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

Definition 11. Let and be and -semihyperrings, respectively. If there exists a map and a bijection such that

for all and , then we say is a homomorphism from to . Also, if is a bijection then is called an isomorphism and and are isomorphic.

Proposition 12. Let and be and -semihyperrings, respectively. Let be a homomorphism from to . If is a -fuzzy sub--semihyperring of , then is a -fuzzy sub--semihyperring of too.

Proof. Suppose that and . Then, we have

Therefore, is a -fuzzy sub--semihyperring of .

Proposition 13. Let and be and -semihyperrings, respectively. Let be a homomorphism from to . If is a -fuzzy -hyperideal of , then is a -fuzzy -hyperideal of too.

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

Let and be two sets of real numbers in . Then, we say is infinitely distributive if

If is continuous, then is infinitely distributive [30].

Lemma 14. Let be a continuous -norm and be a family of -fuzzy sub--semihyperring of . Then, is a -fuzzy sub--semihyperring of .

Proof. For any and , we have

Lemma 15. Let and be and -semihyperrings, respectively, and be an onto homomorphism from to . Then, for every , we have .

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

Proposition 16. Let and be and -semihyperrings, respectively, and let be a fuzzy subset of . Let be an onto homomorphism from to . If is a Min-fuzzy sub--semihyperring of , then is a Min-fuzzy sub--semihyperring of too.

Proof. Suppose that is a Max-fuzzy sub--semihyperring of . By Corollary 8, is a Max-fuzzy sub--semihyperring of if every nonempty level subset is a sub--semihyperring of . Thus, assume that is any nonempty level subset. If , then , and if , then by Lemma 15, we have . By Corollary 8, for each is a sub--semihyperring of . Hence, is a sub--semihyperring of . By Lemma 14, being an intersection of a family of sub--semihyperrings is also a sub--semihyperring of and the proof is completed.

Definition 17. Let and be two -semihyperrings and let and be fuzzy subsets of and , respectively. The product of and is defined to be the fuzzy subset of with , for all .

Proposition 18. Let and be two -semihyperrings and let and be fuzzy subsets of and , respectively. Then

(1)if and are -fuzzy sub- -semihyperrings of and , respectively, then is a -fuzzy sub- -semihyperring of ; (2)if and are -fuzzy -hyperideals of and , respectively, then is a -fuzzy -hyperideal of .

Proof. It is straightforward.

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

Proposition 19 (see [12]). Let be a -semihyperring. Then, the following conditions are equivalent:

(1) satisfying the Acc condition on right (left) -hyperideals, (2)every nonempty family of right (left) -hyperideals has a maximal element, (3)every right (left) -hyperideals is finitely generated.

Definition 20 (see [12]). A -semihyperring is right (left) Noetherian if the equivalent conditions of the above proposition are satisfied. In the same way, we can define an Artinian -semihyperring. Let be a -hyperideal of a -semihyperring and be a Noetherian -semihyperring. Then, is called a Noetherian -hyperideal of .

Example 21 (see [12]). Let for every , , and . Then, is a Noetherian -semihyperring with respect to the following hyperoperations:

where and .

Theorem 22. Let be a family of -hyperideals of a -semihyperring , where . Let be a fuzzy subset of defined by

for all , where stands for . Let be a t-norm with  . Then, is a -fuzzy -hyperideal of .

Proof. Let . Suppose that and for and . Without loss of generality we may assume that . Then, obviously . Since is a -hyperideal of , it follows that and which imply that and for all . If and , then . Hence, and . If and , then there exists such that . It follows that which implies that and for all . Finally, assume that and , then for some . Therefore, , and thus and , for all . Hence, is a -fuzzy -hyperideal of .

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

Proof. Let be a strictly increasing sequence of elements of . Then . Then, is an ideal of for all Let . Then , and so . Hence . Since , there exists such that . It follows that but . Thus, and so we obtain a strictly descending sequence of -hyperideals of which is not terminating. This contradicts the assumption that satisfies the descending chain condition. Consequently, has finite number of values.

Theorem 24. Let be a -semiring, be a fuzzy subset of , and be a -norm with  . Then, the following conditions are equivalent:

(1) is a Noetherian -semihyperring,(2)the set of values of any -hyperideal of is a well-ordered subset of .

Proof. : Let be a -fuzzy -hyperideal of . Suppose that the set of values of is not a well-ordered subset of . Then, there exists a strictly decreasing sequence such that . It follows that is a strictly ascending chain of -hyperideals of , where , for every . This contradicts the assumption that is a Noetherian -semihyperring.

: Suppose that the condition is satisfied and is not a Noetherian -hyperring. There exists a strictly ascending chain

of -hyperideals of . Note that is a -hyperideal of . Define a fuzzy subset in by

We claim that is a -fuzzy -hyperideal of . Let . If and , then and . It follows that and , for all . Suppose that and for all . Since is a -hyperideal of , it follows that . Hence, and so . Similarly, for the case and , we have and , for all . Thus, is a -fuzzy -hyperideal of . Since the chain is not terminating, has a strictly descending sequence of values. This contradicts the assumption that the value set of any ideal is well-ordered. Therefore, is a Noetherian -semihyperring.

For a family of fuzzy subsets in , we define the join and the meet as follows:

for all , where is any index set.

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

Proof. Since is a completely distributive lattice with respect to the usual ordering in , it is sufficient to show that and are -fuzzy -hyperideals of for family of -fuzzy -hyperideals of . For any , we have

Now, let and . Then

Hence, and are -fuzzy -hyperideals of . This completes the proof.

Conflict of Interests

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

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