Abstract and Applied Analysis

Volume 2013 (2013), Article ID 560698, 9 pages

http://dx.doi.org/10.1155/2013/560698

Research Article

Structure of Intuitionistic Fuzzy Sets in -Semihyperrings

Bayram Ali Ersoy¹ and Bijan Davvaz²

¹Department of Mathematics, Yildiz Technical University, 34210 Istanbul, Turkey

²Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran

Received 21 October 2012; Accepted 27 December 2012

Academic Editor: Feyzi Başar

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.

Abstract.

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

1. Introduction

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

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

In 1964, Nobusawa introduced -rings as a generalization of ternary rings. Barnes [12] weakened slightly the conditions in the definition of -ring in the sense of Nobusawa. Barnes [12], Luh [13], and Kyuno [14] studied the structure of -rings and obtained various generalization analogous to corresponding parts in ring theory. The concept of -semigroups was introduced by Sen and Saha [15, 16] as a generalization of semigroups and ternary semigroups. Then the notion of -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 -semihypergroup as a generalization of a semihypergroup. Many classical notions of semigroups and semihypergroups have been extended to -semihypergroups and a lot of results on -semihypergroups are obtained. In [26–29], Davvaz et al. studied the notion of a -semihyperring as a generalization of semiring, a generalization of a semihyperring, and a generalization of a -semiring.

The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures, see [20]. In [30], Davvaz introduced the notion of fuzzy subhypergroups as a generalization of fuzzy subgroups, and this topic was continued by himself and others. In [31], Leoreanu-Fotea and Davvaz studied fuzzy hyperrings. Recently, Davvaz et al. [32–35] considered the intuitionistic fuzzification of the concept of algebraic hyperstructurs and investigated some properties of such hyperstructures.

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

2. Basic Definitions

Let be a nonempty set, and let be the set of all non-empty subsets of . A hyperoperation on is a map , and the couple is called a hypergrupoid. If and are non-empty subsets of , then we denote

A hypergrupoid is called a semihypergroup if for all of we have . That is,

A semihypergroup is called a hypergroup if for all , .

A semihyperring is an algebraic structure which satisfies the following properties:

is a commutative semihypergroup; that is,

(i),

(ii),  for all . is a semihypergroup; that is, , for all . The multiplication is distributive with respect to hyperoperation +; that is,

for all . The element is an absorbing element; that is, , for all . A semihyperring is called commutative if for all . Vougiouklis in [22] studied the notion of semihyperrings in a general form. That is, both sum and multiplication are hyperoperations.

A semihyperring has identity element if there exists , such that , for all . An element is called unit if there exists , such that . A non-empty subset of a semihyperring is called subsemihyperring if and , for all . A left hyperideal of a semihyperring is a non-empty subset of satisfying the following:

(i), for all ,(ii), for all and .

Let and be two non-empty sets. Then, is called a -semihypergroup; if for every hyperoperation , , and , we have

For example, and . For all and for all , we define . Then, is a -semihypergroup.

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

Definition 1. Let be a commutative semihypergroup, and be a commutative group. Then, is called a -semihyperring if there exists a map (image to denoted by satisfying the following conditions:

(i), (ii), (iii), (iv),  for all and .

In the above definition, if is a semigroup, then is called a multiplicative -semihyperring. A -semihyperring is called commutative if , for all and . We say that a -semihyperring is with zero, if there exists , such that and , for all and . Let and be two non-empty subsets of a -semihyperring and . We define

A non-empty subset of of -semihyperring is called a semihyperring if it is closed with respect to the multiplication and addition. In other words, a non-empty subset of -semihyperring is a sub -semihyperring if

Example 2. Let be a semiring, and let be a subsemiring of . We define

the ideal generated by , for all and . Then, it is not difficult to see that is a multiplicative -semihyperring.

Example 3. Let , and . We define for every and . Then, is a -semihyperring under ordinary addition and multiplication.

Example 4. Let be a semihyperring, and let the set of all matrices with entries in . We define by

for all and .

Then, is -semihyperring [29]. Now, suppose that

Now, it is easy to see that is a sub -semihyperring of .

A non-empty subset of a -semihyperring is a left (right) -hyperideal of if for any implies and and is a -hyperideal of if it is both left and right -hyperideal.

Example 5. Consider the following:

Then, is a -semihyperring under the matrix addition and the hyperoperation as follows:

for all and . Let

Then, is a right -hyperideal of , but not left. is a left -hyperideal of , but not right. is both right and left -hyperideal of .

A fuzzy subset of a non-empty set is a function from to . For all , the complement of is the fuzzy subset defined by . The intersection and the union of two fuzzy subsets and of , denoted by and , are defined by

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

is called a fuzzy left -hyperideal of if

is called a fuzzy right -hyperideal of if

is called a fuzzy -hyperideal of if is both a fuzzy left -hyperideal and fuzzy right -hyperideal of .

Example 7. Let , , , and be non-empty subsets of . We define , for every and . Then, is a -semihypering. Now, we define the fuzzy subset of as follows:

It is easy to see that is a fuzzy -hyperideal of .

3. Atanassov’s Intuitionistic Fuzzy -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 in a non-empty set has the form . Here, is the degree of membership of the element to the set , and is the degree of nonmembership of the element to the set . We have also , for all .

Example 8 (see [35]). Consider the universe . An intuitionistic fuzzy set “Large” of denoted by and may be defined by

One may define an intuitionistic fuzzy set “Very Large” denoted by , as follows:

for all . Then,

For the sake of simplicity, we shall use the symbol instead of . Let and be two intuitionistic fuzzy sets of . Then, the following expressions are defined in [4] as follows:

if and only if and , for all ,,,,,.

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

Definition 9. Let be a -semihyperring.

An intuitionistic fuzzy set in is called a left intuitionistic fuzzy -hyperideal of -semihyperring if

(i)  and ,(ii) and , for all .

An intuitionistic fuzzy set in is called a right intuitionistic fuzzy -hyperideal of -semihyperring if

(i)  and ,

(ii) and  for all . An intuitionistic fuzzy set in is called an intuitionistic fuzzy -hyperideal of -semihyperring if it is both left intuitionistic fuzzy -hyperideal and right intuitionistic fuzzy -hyperideal of .

Example 10. Let be a -semihyperring in Example 7. We define the fuzzy subsets and of as follows:

Then, is an intuitionistic fuzzy -hyperideal of .

Example 11. Let be a -semihyperring defined in Example 5. Suppose that

for all . Then, is an intuitionistic fuzzy -hyperideal of .

Theorem 12. Let be an intuitionistic fuzzy -hyperideal of -semihyperring and . We define an intuitionistic fuzzy set in by and , for all . Then, is an intuitionistic fuzzy -hyperideal of .

Proof. Note that .

Theorem 13. Let be an intuitionistic fuzzy -hyperideal of -semihyperring . We define an intuitionistic fuzzy set in by and , for all . Then, is an intuitionistic fuzzy -hyperideals of .

Proof. Since is an intuitionistic fuzzy -hyperideal, we have and so

Hence, ; that is,

Since , so which implies that . Hence, . Also, we have

Moreover, we have

We proved the theorem for left intuitionistic fuzzy -hyperideals. For the proof of right intuitionistic fuzzy -hyperideals similar proof is used.

Lemma 14. If is a collection of intuitionistic fuzzy -hyperideals of , then and are intuitionistic fuzzy -hyperideals of , too.

Proof. The proof is straightforward.

Theorem 15. An intuitionistic fuzzy subset of a -semihyperring is a left (res. right) intuitionistic fuzzy -hyperideal of if and only if for every , the fuzzy sets and are left (res. right) fuzzy -hyperideals of .

Proof. Assume that is a left intuitionistic fuzzy -hyperideal of . Clearly, by using the definition, we have that is a left (res. right) fuzzy -hyperideal.

Also,

for all and .

Conversely, suppose that and are left (res. right) fuzzy -hyperideals of . We have and , for all and . We obtain

Therefore, is a left intuitionistic fuzzy -hyperideal of .

For any fuzzy set of and any , is called an upper bound -of , and is called a lower bound -of .

Theorem 16. An intuitionistic fuzzy subset of a -semihyperring is a left (right) intuitionistic fuzzy -hyperideal of if and only if for every , the subsets and of are left (res. right) -hyperideals, when they are non-empty.

Proof. Assume that is a left intuitionistic fuzzy -hyperideal of . Let . Since , then we have . Let and . We have . Therefore, and so .

Now, let . Since and , we have . Then, , and so . Now, for all , and we have . Therefore, and so .

Conversely, suppose that all non-empty level sets   and are left -hyperideals. Let and . Let , and , with , , then and . Since

we have

Since and are left -hyperideals, and we have and , we have and . And so and . Hence, is a left intuitionistic fuzzy -hyperideal of .

Corollary 17. Let be a left (res. right) -hyperideal of a -semihyperring . We define fuzzy sets and as follows:

where and for . Then, is a left intuitionistic fuzzy -hyperideal of and .

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

Theorem 18. If is an intuitionistic fuzzy -hyperideal of , then so are and .

Proof. Since is an intuitionistic fuzzy -hyperideal of , we have

for all and . Moreover,

This shows that is an intuitionistic fuzzy -hyperideal of . For the other part one can use the similar way.

Note that if we have a fuzzy -hyperideal , then . While for a proper intuitionistic fuzzy -hyperideal , we have and .

Theorem 19. If is a left intuitionistic fuzzy -hyperideal of , then we have,

for all .

Proof. The proof is straightforward.

Let and be two intuitionistic fuzzy -hyperideals of . Then, the product of and denoted by is defined as follows:

where

Let be the characteristic function of a left (res. right) -hyperideal of . Then, is a left (res. right) intuitionistic fuzzy -hyperideal of .

Theorem 20. If is a -semihyperring and is an intuitionistic fuzzy -hyperideal of , then , where and is the characteristic function of .

Proof. The proof is straightforward.

Proposition 21. Let be a -semihyperring, and let be an intuitionistic fuzzy -hyperideal of . Then,

if is a fixed element of , then the set and are -hyperideals of , the sets and are -hyperideals of .

Proof. (1) Suppose that . Then, we have

which implies that and so . Similarly, suppose that . We have which implies that and so . Assume that , and . Then, we have which implies that and so . Similarly, suppose that , and . We have which implies that and so .

(2) One can easily prove the second part.

Definition 22. Let be a -semihypering, and let be a -semihyperring. 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.

Definition 23. Let and be two non-empty intuitionistic fuzzy subsets of and , respectively, and let be a map. Then,

(1)the inverse image of under is defined by

where

(2)the image of under , denoted by , is the intuitionistic fuzzy set in defined by , where for each

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

Theorem 24. Let be a -semihyperring, let be a -semihyperring, and let be a homomorphism from to . Then,

if is an intuitionistic fuzzy -hyperideal of , then is an intuitionistic fuzzy -hyperideal of, if is an intuitionistic fuzzy -hyperideal of, then is an intuitionistic fuzzy -hyperideal of.

Proof. The proof is straightforward.

4. Artinian and Noetherian -Semihypergroups

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

Definition 25. Let be a -semihypering. Then, is called Noetherian (Artinian resp.) if satisfies the ascending (descending) chain condition on -hyperideals. That is, for any -hyperideals of , such that

there exists , such that , for all .

Proposition 26 (see [28]). Let be a -semihypering. Then, the following conditions are equivalent:

is Noetherian. satisfies the maximum condition for -hyperideals. Every -hyperideal of is finitely generated.

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

Proof. Suppose that every intuitionistic fuzzy -hyperideal of -semihyperings has finite number of values, and is not Artinian. So, there exists a strictly descending chain

of -hyperideals of . We define the intuitionistic fuzzy set by

It is easy to see that is an intuitionistic fuzzy -hyperideal of . We have contradiction because of the definition of , which depends on infinitely descending chain of -hyperideals of .

Theorem 28. Let be a -semihypering. Then the following statements are equivalent:

is Noetherian. The set of values of any intuitionistic fuzzy -hyperideal of is a well-ordered subset of .

Proof. Let be an intuitionistic fuzzy -hyperideal of -semihypering. Assume that the set of values of is not a well-ordered subset of . Then, there exits a strictly infinite decreasing sequence , such that and . Let and . Then, and are strictly infinite decreasing chains of -hyperideals of . This contradicts with our hypothesis.

Suppose that is a strictly infinite ascending chain -hyperideals of . Let . It is easy to see that is a -hyperideal of . Now, we define

Clearly, is an intuitionistic fuzzy -hyperideal of . Since the chain is not finite, has strictly infinite ascending sequence of values. This contradicts that the set of the values of fuzzy -hyperideal is not well ordered.

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

Proof. Suppose that is an intuitionistic fuzzy -hyperideal of , and are not finite. According the previous theorem, we consider the following two cases.

Case 1. Suppose that is strictly increasing sequence in and is strictly decreasing sequence in . Now, we obtain

are strictly descending and ascending chains of-hyperideals, respectively. Since is both Artinian and Noetherian there exists a natural number , such that and for . This means that and . This is a contradiction.

Case 2. Assume that , are strictly decreasing and increasing sequence in and , respectively. From these equalities we conclude that

are strictly ascending and descending chains of -hyperideals, respectively. Because of the hypothesis there exists a natural number , such that and for . This implies that and 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.

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