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Journal of Applied Mathematics
Volume 2014 (2014), Article ID 173267, 9 pages
http://dx.doi.org/10.1155/2014/173267
Research Article

Some Generalized Forms of Fuzzy Interval Valued Hyperideals in a Hyperring

1Mathematics Department, Zhoukou Normal University, Zhoukou 466001, China
2College of Mechanical and Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China
3College of Sciences, East China Institute of Technology, Nanchang 330013, China
4Faculty of Science, Kunming University of Science and Technology, Kunming 650093, China

Received 15 October 2013; Revised 20 May 2014; Accepted 22 May 2014; Published 12 June 2014

Academic Editor: Hector Pomares

Copyright © 2014 Hongjie Li 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

Some generalized forms of the hyperideals of a hyperring in the paper of Zhan et al. (2008) will be given. As a generalization of the interval valued -fuzzy hyperideals of a hyperring with and , the notion of generalized interval valued -fuzzy hyperideals of a hyperring is also introduced. Special attention is concentrated on the interval valued -fuzzy hyperideals. As a consequence, some characterizations theorems of interval valued -fuzzy hyperideals will be provided.

1. Introduction

The concept of hyperstructure was first introduced by Marty [1] in 1934. People later observed that hyperstructures have many applications in pure and applied sciences. A comprehensive review of hyperstructures can be found in [2, 3]. In a recent monograph of Corsini and Leoreanu [4], the authors have collected numerous applications of algebraic hyperstructures, especially those from the last decade to the following subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and rough sets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities.

After introducing the concept of fuzzy sets of Zadeh in 1965 (see [5]), there are many papers devoted to fuzzify the classical mathematics into fuzzy mathematics. The relationships between the fuzzy sets and algebraic hyperstructures (structures) have been considered by Corsini, Davvaz, Kehagias, Leoreanu, Yin, Zahedi, Zhan, and others. The reader is referred to the papers [624]. By using the notion “belongingness ()” and “quasicoincidence ” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], the concept of -fuzzy subgroups, where , are any two in with , was introduced by Bhakat and Das [26] in 1992, in which they first generalized Rosenfeld’s fuzzy subgroup [27]. The detailed study of -fuzzy subgroups has been considered by Bhakat and Das in [28] and Bhakat in [29, 30]. The concept of the -fuzzy subhyperquasigroups of hyperquasigroups was introduced by Davvaz and Corsini [12]. By using the idea of “quasicoincidence” of a fuzzy interval value with an interval valued fuzzy set, Zhan et al. [21] introduced and investigated the notion of interval valued -fuzzy hyperideals of a hyperring.

As a generalization of the concepts of “belongingness ()” and “quasicoincidence ” of a fuzzy point with a fuzzy set introduced by Pu and Liu [25], Yin and Zhan [31] introduced the concept of “-belongingness ()” and “-quasicoincidence ()” of a fuzzy point with a fuzzy set. Using these new concepts, Yin and Zhan [31] introduced the concepts of -fuzzy (implicative, positive implicative, and fantastic) filters and -fuzzy (implicative, positive implicative, and fantastic) filters of BL-algebras, where , , , and , and some related properties were investigated. This idea is continued and studied by Ma et al. [32], Ma and Zhan [33], Yin and Zhan [34], and so on. Following this idea, in this paper, we will give some of the generalized forms of the interval fuzzy hyperideals given by Zhan et al. in [21]. The notion of generalized interval valued -fuzzy hyperideals is introduced which is the generalization of the notion of interval valued -fuzzy hyperideals. Special attention will be paid to the interval valued -fuzzy hyperideals. It is noteworthy that the notion of an interval valued -fuzzy hyperideal is a special generalized interval valued -fuzzy hyperideal and hence many results in [21] will become easy corollaries of our results given in this paper.

2. Hyperrings

We first recall that a hyperstructure is a nonempty set together with a mapping “”: , where is the set of all nonempty subsets of , written as . We note that if and are nonempty subsets of , then by and , we mean that and , respectively.

Now, we call a hyperstructure a canonical hypergroup [35] if the following axioms are satisfied:(1)for every , ;(2)for every , ;(3)there is a unique such that , for all ;(4)for every , there exists a unique element such that (we call the element the opposite of ).

Definition 1 (see [36]). A hyperring is an algebraic structure which satisfies the following axioms.(1) is a canonical hypergroup (we will write for ).(2) is a semigroup having zero as a bilaterally absorbing element.(3)The multiplication is distributive with respect to the hyperoperation “+.”

Let be a hyperring and let be a nonempty subset of . Then is called a subhyperring of if itself is a hyperring. In what follows, let denote a hyperring unless otherwise stated. subhyperring of is a left (right) hyperideal of if , for all and . subhyperring is called a hyperideal if is both left and right hyperideal.

3. Interval Valued Fuzzy Sets

By an interval number [37], we mean an interval , where . The set of all interval numbers is denoted by . The interval can be simply identified by the number . For the interval numbers , , where is an index set, we define

For any intervals and , where and , we define and we put(i) and ,(ii) and ,(iii) and .Denote by the set all of interval numbers such that any two elements of are comparable. Then, it is clear that both and form a complete lattice with as their least element and as their greatest element.

Recall that an interval valued fuzzy subset on is the set where and are two fuzzy subsets of such that for all . Putting , we see that , where . Denote by the set of all interval valued fuzzy subsets of such that, for any , Im.

If and are two interval valued fuzzy subsets of , then we define

if and only if, for all , and ,

if and only if, for all , and .

Also, the union and intersection are defined as follows:

An interval valued fuzzy subset of of the form is said to be a fuzzy interval value with support and interval value and is denoted by .

Let be such that . For a fuzzy interval value and an interval valued fuzzy set , we say that(1) if ;(2) if ;(3) if or ;(4) if and ;(5) if does not hold for .

Next we define a new ordering relation “” on , which is called the interval valued fuzzy inclusion or quasicoincidence relation, as follows.

For any and such that , by we mean that for all and .

And we define a relation “” on as follows.

For any , by we mean that and .

In the sequel, unless otherwise stated, we let be a hyperring, , such that and denote by that is not true. We also emphasize that any interval valued fuzzy subset of must satisfy the following conditions:

or and or for all .

Lemma 2. Let be a nonempty set and . Then if and only if for all .

Proof. Assume that . Let . If , then there exists such that ; that is, but , a contradiction. Hence .
Conversely, assume that for all . If , then there exist and such that but , and so , , and . This gives that , a contradiction. Hence .

Lemma 3. Let be a nonempty set and such that . Then .

Proof. It is straightforward by Lemma 2.

Note that Lemma 2 gives that if and only if for all and , and it follows from Lemmas 2 and 3 that “” is an equivalence relation on .

We now define some operations of interval valued fuzzy subsets of .

Definition 4. Let . Define the sum and product of and , denoted by and , by for all .

By Lemma 2 and Definition 4, we can easily deduce the following results.

Lemma 5. Let be such that and . Then(1), ;(2).

Lemma 6. Let . Consider(1);(2);(3);(4).

Lemma 5 indicates that the equivalence relation “” is a congruence relation on .

4. Generalized Interval Valued -Fuzzy Hyperideals

In this section, we define and investigate generalized interval valued -fuzzy hyperideals of a hyperring, where and .

Definition 7. An interval valued fuzzy subset of is called an interval valued -fuzzy hyperideal of if, for all , , and :(F1a) and for all ;(F2a) for all ;(F3a) and for all .

An interval valued -fuzzy hyperideal with and is called an interval valued -fuzzy hyperideal studied in [21].

The case must be omitted since for an interval valued fuzzy subset of such that for any in the case we have and . Thus , which implies . This means that .

As it is not difficult to see, each interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of . So, in the theory of interval valued -fuzzy hyperideals the central role is played by interval valued -fuzzy hyperideals and we only need to investigate the properties of interval valued -fuzzy hyperideals. In what follows, let unless otherwise specified.

Example 8. Let be a set with a hyperoperation “+” and a binary operation “” as follows: Then is a hyperring ([21]). Define an interval valued fuzzy subset of by Then is an interval valued -fuzzy hyperideal of .

Proposition 9. An interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of .

Proof. This is straightforward since implies for all and .

Proposition 10. Any interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of .

Proof. This part is straightforward.

The following example shows that the converse of Propositions 9 and 10 may not be true in general.

Example 11. Consider Example 8. Define an interval valued fuzzy subset of by Then(1)it is routine to verify that is an -fuzzy hyperideal of ;(2) is not an -fuzzy hyperideal of since but and ;(3) is not an -fuzzy hyperideal of since and but .

Theorem 12. Let and be an interval valued -fuzzy hyperideal of . Then the set is a hyperideal of , where .

Proof. Assume that is an interval valued -fuzzy hyperideal of . Let . Then , . We consider the following two cases.
Case 1. If , then and . By (F1a), for all , that is, or . It follows that or , and so or . Hence and so . Therefore, .
By (F2a) and (F3a), and . Analogous to the above proof, we have and ; that is, and . Therefore, is a hyperideal of .
Case 2. If , then and since . Analogous to the proof of Case 1, we may prove that is a hyperideal of .

If we take and in Theorem 12, then we have the following corollary.

Corollary 13. Let be an interval valued -fuzzy hyperideal of . Then the set is a hyperideal of , where .

Theorem 14. Let and be a nonempty subset of . Then is a hyperideal of if and only if the interval valued fuzzy subset of such that for all and otherwise is an interval valued -fuzzy hyperideal of .

Proof. Assume that is a hyperideal of . Let and be such that and . Then we have the following four cases.
Case 1. If and , then and . Thus, and ; that is, .
Case 2. If and , then and , and so and . It follows that and ; that is, .
Case 3. If and , then and . Analogous to the proof of Cases 1 and 2, and ; that is, .
Case 4. If and , then and . Analogous to the proof of Cases 1 and 2, and ; that is, .
Thus, in any case, . Hence for all , which implies that . If , then ; it follows that . If , then ; it follows that . Therefore, . In a similar way, we may prove that and . Therefore, is an interval valued -fuzzy hyperideal of .
Conversely, assume that is an interval valued -fuzzy hyperideal of . It is easy to see that . It follows from Theorem 12 that is a hyperideal of .

If we take and in Theorem 14, then we have the following corollary.

Corollary 15. Let be a nonempty subset of . Then is a hyperideal of if and only if the interval valued fuzzy subset of such that for all and otherwise is an -fuzzy hyperideal of , where .

Theorem 16. Let be an -fuzzy hyperideal of . Then the following conditions hold:(F1b) for all ;(F2b) for all ;(F3b) for all .

Proof. Let be an -fuzzy hyperideal of . We only show (F1b). The other properties can be similarly proved. Assume that there exist such that and . Then, for all such that , we have and so , , , and . Hence , but , a contradiction. Hence (F1b) is valid.

5. Interval Valued -Fuzzy Hyperideals

In this section, using a new ordering relation on , we investigate the interval valued -fuzzy hyperideals of a hyperring. In what follows, we take and in Definition 7. Before proceeding, we present some characterizations of interval valued -fuzzy hyperideals.

Lemma 17. Let . Then (F1a) holds if and only if one of the following conditions holds:(F1b) for all ;(F1c).

Proof. (F1a)(F1b) Let . Let, if possible, be such that and . Then , , , and ; that is, , a contradiction. Hence (F1b) is valid.
(F1b)(F1c) For , suppose, if possible, . Then and ; that is, and which implies that . If for some , by (F1b), we have , it follows that . Hence we have which contradicts and . Hence (F1c) is satisfied.
(F1c)(F1a) Let and be such that and . Then for any , we have Hence and so (F1a) holds.

Lemma 18. Let . Then (F2a) holds if and only if one of the following conditions holds:(F2b) for all ;(F2c), where is defined by for all .

Proof. The proof is analogous to the proof of Lemma 17.

Lemma 19. Let . Then (F3a) holds if and only if one of the following conditions holds:(F3b) for all ;(F3c).

Proof. The proof is analogous to Lemma 17.

From Lemmas 1719, we obtain the following results.

Theorem 20. Let . Then is an -fuzzy hyperideal of if and only if it satisfies conditions (F1b)–(F3b) or conditions (F1c)–(F3c).

Remark 21. For any interval valued -fuzzy hyperideal of , we can conclude that(1) is an interval valued fuzzy hyperideal of when and ([21], Definition 4.1);(2) is an interval valued -fuzzy hyperideal of when and ([21], Definition 4.3);(3) is an interval valued fuzzy hyperideal of with thresholds ([21], Definition 4.9).

Combining Theorems 16 and 20, we have the following result.

Proposition 22. Any interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of .

The following example shows that the converse of Proposition 22 is not true in general.

Example 23. Let and be as in Example 11. Then is an -fuzzy hyperideal of , but it is not an -fuzzy hyperideal of since and but .

If we take and in Propositions 9, 10, and 22, then we have the following corollary.

Corollary 24. Let . Then(1)any interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of [21];(2)any interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of ;(3)any interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of [21].

For any , we define , , and for all . It is clear that .

The next theorem provides the relationship between the interval valued -fuzzy hyperideals of and the crisp hyperideals of .

Theorem 25. Let .(1) is an interval valued -fuzzy hyperideal of if and only if is an hyperideal of for all .(2)If , then is an interval valued -fuzzy hyperideal of if and only if is a hyperideal of for all .(3)If , then is an interval valued -fuzzy hyperideal of if and only if is a hyperideal of for all .

Proof. We only show (3). Let be an interval valued -fuzzy hyperideal of and for some . Then and ; that is, or , and or . Since is an interval valued -fuzzy hyperideal of , we have for all since . We consider the following cases.
Case 1. Consider . Since , we have and so or , or . Hence .
Case 2. Consider . Since , we have and so or , or . Hence .
Thus, in any case, ; that is, for all . Similarly we can show that and . Therefore, is a hyperideal of .
Conversely, assume that the given condition holds. Let . If there exists such that and , then but , a contradiction. Therefore, for all . Similarly we can show that conditions (F2b) and (F3b) are valid. Therefore, is an interval valued -fuzzy hyperideal of by Theorem 20.

As a direct consequence of Theorem 25, we have the following result.

Corollary 26. Let be such that , , , and . Then every interval valued -fuzzy hyperideal of is an interval valued -fuzzy hyperideal of .

The following example shows that the converse of Corollary 26 is not true in general.

Example 27. Consider Example 8. Define interval valued fuzzy subset of by Then is an interval valued -fuzzy hyperideal of but is not an interval valued -fuzzy hyperideal of .

If we take and in Theorem 25, then we have the following corollary.

Corollary 28. Let .(1) is an interval valued -fuzzy hyperideal of if and only if is a hyperideal of for all .(2) is an interval valued -fuzzy hyperideal of if and only if is a hyperideal of for all .(3) is an interval valued -fuzzy hyperideal of if and only if is a hyperideal of for all .

Theorem 29. Let and be two interval valued -fuzzy hyperideals of . Then so are and .

Proof. We only show that is an interval valued -fuzzy hyperideal of . The verification is as follows.(1)By Lemmas 5, 6, and 17, .(2)Let . Then (3)Let . We have In a similar way, we have . Hence, .

Theorem 30. Let be the set of all interval valued -fuzzy hyperideals of with the same tip “i.e., for all . Then is a modular lattice.

Proof. Let . Clearly, is the greatest lower bound of and . By Theorem 29, we have . Now, we prove that . Let . Since and for all , we have it follows that . Similarly, and so is an upper bound of and . Now, let be such that and . It is easy to see . So, . Therefore, is a lattice.
For modularity, we first let be such that . We need to show that . But, the relation is clear. It remains to show that . Now let . Then we have