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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 348203, 13 pages

http://dx.doi.org/10.1155/2013/348203

## Characterizations of Semihyperrings by Their ()-Fuzzy Hyperideals

^{1}College of Mathematical Sciences, Honghe University, Mengzi 661199, China^{2}School of Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China^{3}Department of Mathematics, Hubei Institute for Nationalities, Enshi, Hubei 445000, China

Received 21 December 2012; Accepted 19 March 2013

Academic Editor: Hector Pomares

Copyright © 2013 Xiaokun Huang 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 concepts of ()-fuzzy bi-hyperideals and ()-fuzzy quasi-hyperideals of a semihyperring are introduced, and some related properties of such ()-fuzzy hyperideals are investigated. In particular, the notions of hyperregular semihyperrings and left duo semihyperrings are given, and their characterizations in terms of hyperideals and ()-fuzzy hyperideals are studied.

#### 1. Introduction

The concept of the fuzzy set, initiated by Zadeh in his pioneer paper [1] of 1965, is an important tool for modeling uncertainties in many complicated problems in engineering, economics, environment, medical science, and social science, due to information incompleteness, randomness, limitations of measuring instruments, and so forth. Many classical mathematics is extended to fuzzy mathematics, and various properties of them in the context of fuzzy sets are established. The idea of quasi-coincidence of a fuzzy point with a fuzzy set, which is mentioned in [2], played a prominent role to generalize some basic concepts of fuzzy algebraic systems. Bhakat and Das [3, 4] gave the concepts of -fuzzy subgroups by using the “belongs to” relation () and “quasi-coincident with” relation between a fuzzy point and a fuzzy subgroup and introduced the concept of -fuzzy subgroup. They also defined and investigated the -fuzzy subrings and ideals of a ring in [5]. Later on, Dudek et al. [6] introduced the notion of -fuzzy ideal and -fuzzy -ideal in hemirings. Davvaz et al. [7] considered the concept of interval-valued -fuzzy -submodules of -modules. Recently, Yin and Zhan [8] further generalized the above relations and by using a pair of thresholds and (). They gave some new relations and between a fuzzy point and a fuzzy set and studied -fuzzy filters in -algebras where are two of with . Afterwards, this direction was continued by Ma and Zhan, and others (for instance, [9, 10]).

Algebraic hyperstructure was introduced in 1934 by a French mathematician, Marty [11], at the 8th Congress of Scandinavian Mathematicians. Later on, people have observed that hyperstructures have many applications in both pure and applied sciences. A comprehensive review of the theory of hyperstructures can be found in [12–14]. In a recent book of Corsini and Leoreanu [15], the authors have collected numerous applications of algebraic hyperstructures, especially those from the last fifteen years to the following subjects: geometry, hypergraphs, binaryrelations, lattices, fuzzy sets and roughsets, automata, cryptography, codes, median algebras, relation algebras, artificial intelligence, and probabilities. The study of fuzzy hyperstructures is also an interesting research topic of hyperstructure. Many fuzzy theorems in hyperstructures have been discussed by several authors, for example, Corsini, Cristea, Davvaz, Kazanci, Leoreanu, Yin, and Zhan (see, e.g., [7, 8, 13, 16–24]). Hyperstructure, in particular semihyperring, is a generalization of classical algebra in which the ordinary operations are replaced by hyperoperations which map a pair of elements into a subset. In [25], Ameri and Hedayati gave the notions of semihyperrings and studied the -hyperideals of them. Davvaz [26] gave the concepts of ternary semihyperrings and investigated their fuzzy hyperideals. Dehkordi and Davvaz introduced the notions of -semihyperrings and discussed roughness and a kind of strong regular (equivalence) relations on -semihyperrings (see, [27, 28]). However, one can see that these semihyperrings are based on a hyperoperation and an ordinary operation (or -operation). In this paper, we consider another kind of semihyperring in which addition and multiplication are both hyperoperations. We define -fuzzy hyperideals, -fuzzy bi-hyperideals, and -fuzzy quasi-hyperideals in a semihyperring and investigate some related properties of them. In the rest, we give the concepts of hyperregular semihyperrings and left duo semihyperrings and address their characterizations in terms of -fuzzy hyperideals, -fuzzy bi-hyperideals, and -fuzzy quasi-hyperideals.

#### 2. Preliminaries

Let be a set and the family of all nonempty subsets of and a hyperoperation or join operation; that is, is a map from to . If , its image under is denoted by . If , then is given by . is used for and for . Generally, the singleton is identified by its element .

together with a hyperoperation is called a semihypergroup if for all . A semihypergroup is said to be commutative if for all . The concept of semihyperrings was introduced by Ameri and Hedayati in 2007 [25]. We formulate it as follows.

*Definition 1 (see [25]). *A semihyperring is an algebraic hyperstructure satisfying the following axioms:(1) is a commutative semihypergroup with a zero element satisfying ;(2) is semigroup; that is, for all ;(3)the multiplication is distributive over the hyperoperation +; that is, for any , we have and ;(4)the element is an absorbing element; that is, for all .

In Definition 1, if the multiplication is replaced by a hyperoperation, then we have the following definition.

*Definition 2. *A semihyperring is an algebraic hyperstructure consisting of a nonempty set together with two binary hyperoperations on satisfying the following axioms:(1) is a commutative semihypergroup;(2) is semihypergroup;(3)the hyperoperation is distributive over the hyperoperation +; that is, for any , we have and ;(4)there exists an element such that and , which is called the zero of .

Let be a semihyperring; for any , we write .

*Example 3. *Let be a set with two hyperoperations and as follows:
Then is a semihyperring with a zero.

A nonempty subset (resp., ) of a semihyperring is called a left (resp., right) hyperideal of if it satisfies and (resp., ). A nonempty subset of a semihyperring is called a bi-hyperideal of if it satisfies , and . A nonempty subset of a semihyperring is called a quasi-hyperideal of if it satisfies and .

A fuzzy subset of a semihyperring , by definition, is an arbitrary mapping , where is the usual interval of real numbers. We denote by the set of all fuzzy subsets of .

In what follows let be such that and a semihyperring with a zero . For any , we define to be the fuzzy subset of by for all and otherwise. Clearly, is the characteristic function of if and .

A fuzzy subset in defined by is said to be a fuzzy point with support and value and is denoted by .

For a fuzzy point and a fuzzy subset of , we say that(1) if ,(2) if ,(3) if or .

In the sequel, unless otherwise stated, means that does not hold, where . For any , by , we mean that implies for all and .

Lemma 4. *Let and be two fuzzy subsets of , and then the following conditions are equivalent.*(1)*.*(2)* for all .*(3)* for all , where . *

*Proof. *(1)(2) Assume that (1) holds. Let and be any fuzzy subsets of . If for some , then there exists such that ; it follows that but , a contradiction. Hence for all .

(2)(1) Assume that (2) holds. If does not hold, then there exists but , which contradicts . Therefore, (1) holds.

It is easy to check that (1)(3). This completes the proof.

According to Lemma 4, it can be easily seen that for all . Also, the following result can be easily deduced.

Lemma 5. *Let , and be any fuzzy subsets of such that and , and then .*

We will write if and . Let be the relation which is defined in the above two fuzzy sets of a hypersemigorup . Then it satisfies reflexivity, symmetry, and transitivity. That is, it is an equivalence relation on the .

For two fuzzy subsets and of a semihyperring ; the sum is defined by and if cannot be expressible as an element in for all .

*Definition 6. *Let and be fuzzy subsets of a semihyperring , the intrinsic product is defined by:
and if cannot be expressible as an element in for all .

Lemma 7. *Let , and be any fuzzy subsets of such that and , and then*(1)*,*(2)*.*

*Proof. *(1) For any , by Lemma 4, we have
This implies that .

(2) It is straightforward by Lemma 4.

Lemma 8. *Let , and be any fuzzy subsets of . Then*(1)*,*(2)*, ,*(3)*, .*

*Proof. *It is straightforward.

Lemma 9. *Let be any subsets in . Then one has*(1)* if and only if ,*(2)*,*(3)*.*

*Proof. *(1) Assume that . If , then there exists but . This implies that and , which contradicts . Conversely, assume that . Let and be such that . Then , and so . Thus and . This gives that . Hence .

(2) It is straightforward.

(3) Let . If , then and for some and . Thus we have
This implies that .

If , then and cannot be expressible as a element in for any and . Thus . This implies that .

*Definition 10. *A fuzzy subset in a semihyperring is called an -fuzzy left (resp., right) hyperideal if it satisfies(F1a),(F2a) (resp., ).

A fuzzy subset of a semihyperring is called an -fuzzy hyperideal if it is both an -fuzzy left hyperideal and an -fuzzy right hyperideal.

*Definition 11. *A fuzzy subset in a semihyperring is called an -fuzzy bi-hyperideal if it satisfies conditions (F1a) and(F3a),(F4a).

*Definition 12. *A fuzzy subset in a semihyperring is called an -fuzzy quasi-hyperideal if it satisfies conditions (F1a) and(F5a).

Theorem 13. *Let be a fuzzy subset of a semihyperring .Then is an -fuzzy left (resp., right) hyperideal if and only if for any , one has*(F1b)* for all .*(F2b)* (resp., ). *

*Proof. *Assume that (F1a) holds. For any , if possible, let . Choose such that . Then there exists such that ; that is, . Then ; hence , which contradicts (F1a). Therefore . That is, (F1b) holds.

Conversely, assume that (F1b) holds. Let and be such that , if possible; let . Then and . Hence . Now, (since and the assumption imply that ), a contradiction. Hence, . This implies that . Therefore (F1a) holds.

In a similar way, we can prove that (F2a)(F2b).

Theorem 14. *Let be a fuzzy subset of a semihyperring . Then is an -fuzzy bi-hyperideal if and only if it satisfies (F1b) and for any , one has*(F3b)*,*(F4b)*.*

*Proof. *The proof is analogous to that of Theorem 13.

*Remark 15. *Let be any fuzzy subsets of . If is an -fuzzy left (resp., right) hyperideal of , then implies that (resp., ). If is an -fuzzy bi-hyperideal of , then implies that .

Theorem 16. *(1) is an -fuzzy quasi-hyperideal of for every -fuzzy left hyperideal and every -fuzzy right hyperideal of .**(2) Any -fuzzy quasi-hyperideal of is an -fuzzy bi-hyperideal. *

*Proof. *(1) Let and be any -fuzzy left hyperideal and any -fuzzy left hyperideal of , respectively. Then and . By Lemma 7, we have , and so is an -fuzzy quasi-hyperideal of .

(2) Let be any -fuzzy quasi-hyperideal of . Then, , , and , and so and . Hence is an -fuzzy bi-hyperideal of .

One may easily see that any -fuzzy left hyperideal and any -fuzzy right hyperideal of a semihyperring are an -fuzzy quasi-hyperideal of . However, the converse of the property and Theorem 16 do not hold in general as shown in the following examples.

*Example 17. *Let be a set with two hyperoperations and as follows:
Then is a semihyperring and is a quasi-hyperideal of , and it is not a left (right) hyperideal of . Define the fuzzy set of as follows:
Then is an -fuzzy quasi-hyperideal of , and it is not an -fuzzy left (right) hyperideal of .

*Example 18. *Let be a set with two hyperoperations and as follows:
Then is a bi-hyperideal of , and it is not a quasi-hyperideal of . Define the fuzzy set of as follows:
Then is an -fuzzy bi-hyperideal of , and it is not an -fuzzy quasi-hyperideal of .

Lemma 19. *Let be any subset in . Then the following conditions hold.*(1)* is a left (resp., right) hyperideal of if and only if is an -fuzzy left (resp., right) hyperideal of .*(2)* is a bi-hyperideal of if and only if is an -fuzzy bi-hyperideal of .*(3)* is a quasi-hyperideal of if and only if is an -fuzzy quasi-hyperideal of . *

*Proof. *Let be any subset in . According to Lemma 9, if and only if . Hence is a left hyperideal of if and only if is an -fuzzy left hyperideal of .

The case for bi-hyperdeals and quasi-hyperideals can be similarly proven.

#### 3. Characterizations of Hyperregular Semihyperrings

*Definition 20. *A semihyperring is said to be hyperregular if for each , there exists such that . Equivalent definitions: (1) for all ; (2) for all .

*Example 21. *Let be a set with two hyperoperations and as follows:
Then is a hyperregular semihyperring. Since , and .

*Remark 22. *If is a hyperregular semihyperring, then and .

Theorem 23. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every -fuzzy right hyperideal and every -fuzzy left hyperideal of .*(3)* for every right hyperideal and every left hyperideal of .*

*Proof. *(1) (2) Let be a hyperregular semihyperring, with being any -fuzzy right hyperideal and any -fuzzy left hyperideal of , respectively. Then and . Hence . Let be any element of . Then, since is hyperregular, there exists such that , and then for some . Now, since is -fuzzy right hyperideal of , by Remark 15, we have . Hence,
This implies that . Hence (2) holds.

(2)(3) It is straightforward by Lemmas 9 and 19.

(3)(1) Let be any element of . Then and , where and are the principal right hyperideal and principal left hyperideal generated by , respectively. By the assumption, we have
This implies that for some . Hence is hyperregular.

Corollary 24. *Let be semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every fuzzy subset and every -fuzzy left hyperideal of .*(3)* for every -fuzzy right hyperideal and every fuzzy subset of . *

Corollary 25. *Let be a hyperregular semihyperring. Then every -fuzzy bi-hyperideal is an -fuzzy quasi-hyperideal of . *

*Proof. *Let be any -fuzzy bi-hyperideal of . Evidently, and are an -fuzzy right hyperideal and an -fuzzy left hyperideal of , respectively. Now, by Theorem 23 and Remark 22, we have
This implies that is an -fuzzy quasi-hyperideal of .

Lemma 26. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every bi-hyperideal of .*(3)* for every quasi-hyperideal of . *

*Proof. *(1)(2) Assume that (1) holds. Let be any bi-hyperideal of and any element of . Then there exists such that . It is easy to see that and so . Hence, . On the other hand, since is an bi-hyperideal of , we have . Therefore, .

(2)(3) Evidently, every quasi-hyperideal of is a bi-hyperideal of . Then by the assumption, we have for every quasi-hyperideal of .

(3)(1) Assume that (3) holds. Let and be any right hyperideal and any left hyperideal of , respectively. Then we have , and so it is easy to see that is a quasi-hyperideal of . By the assumption and Theorem 23, we have . Hence, and so is hyperregular.

Theorem 27. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every -fuzzy bi-hyperideal of .*(3)* for every -fuzzy quasi-hyperideal of . *

*Proof. *(1)(2) Assume that (1) holds. Let be any -fuzzy bi-hyperideal of and any element of . Then . On the other hand, since is hyperregular, there exists such that , and then for some . Thus, by Remark 15, we have
This implies that . Hence we have .

(2)(3) It is straightforward by Theorem 16.

(3)(1) Assume that (3) holds. Let be any quasi-hyperideal of . Then by Lemma 19, is an -fuzzy quasi-hyperideal of . Then, by the assumption and Lemma 9, we have
It follows from Lemma 9 that . Now, since is a quasi-hyperideal of , we have , and so . Therefore is hyperregular by Lemma 26.

Corollary 28. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every fuzzy subset of . *

Theorem 29. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every -fuzzy bi-hyperideal and every -fuzzy hyperideal of .*(3)* for every -fuzzy quasi-hyperideal and every -fuzzy hyperideal of . *

*Proof. *(1)(2) Assume that (1) holds. Let and be any -fuzzy bi-hyperideal and any -fuzzy hyperideal of , respectively. Then it is clear that . Now let be any element of . Then since is hyperregular, there exists such that . Then there exist , such that and . Now, since is an -fuzzy hyperideal of , by Remark 15, we have . Thus,
This implies that . Hence, (2) holds.

(2)(3) This is straightforward by Theorem 16.

(3)(1) Assume that (3) holds. Let be any fuzzy quasi-hyperideal of . Then since is an -fuzzy hyperideal of , we have
Then it follows from Theorem 27 that is hyperregular.

Corollary 30. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every bi-hyperideal and every hyperideal of .*(3)* for every quasi-hyperideal and every hyperideal of . *

*Proof. *(1)(2) Assume that (1) holds. Let and be any bi-hyperideal and any hyperideal of , respectively. Then by Lemma 19, and are an -fuzzy bi-hyperideal and an -fuzzy hyperideal of , respectively. Thus, by Theorem 29, we have
Then it follows from Lemma 9 that .

(2)(3) It is straightforward.

(3)(1) Assume that (3) holds. Let be any quasi-hyperideal of . Then since itself is a hyperideal of , we have
Then it follows from Lemma 26 that is hyperregular.

Theorem 31. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every -fuzzy bi-hyperideal and every -fuzzy left hyperideal of .*(3)* for every -fuzzy quasi-hyperideal and every -fuzzy left hyperideal of .*(4)* for every -fuzzy right hyperideal and every -fuzzy bi-hyperideal of .*(5)* for every -fuzzy right hyperideal and every -fuzzy quasi-hyperideal of .*(6)* for every -fuzzy right hyperideal , every -fuzzy bi-hyperideal , and every -fuzzy left hyperideal of .*(7)* for every -fuzzy right hyperideal , every -fuzzy quasi-hyperideal , and every -fuzzy left hyperideal of . *

*Proof. *(1)(2) Assume that (1) holds. Let and be any -fuzzy bi-hyperideal and any -fuzzy left hyperideal of , respectively. Now let be any element of . Since is hyperregular, there exists such that , and then for some . Since is an -fuzzy left hyperideal of , we have . Thus,
This implies that .

(2)(1) Assume that (2) holds. Let and be any -fuzzy right hyperideal and any -fuzzy left hyperideal of , respectively. Then it is easy to check that is an -fuzzy bi-hyperideal of . By the assumption, we have . Hence, . So is hyperregular by Theorem 23.

Similarly, we can show that (1)(3), (1)(4), and (1)(5).

(1)(6) Assume that (1) holds. Let , , and be any -fuzzy right hyperideal, any -fuzzy bi-hyperideal, and any -fuzzy left hyperideal of , respectively. Now let be any element of . Since is hyperregular, there exists such that . Then there exist , and such that . Now, since is an -fuzzy right hyperideal and is an -fuzzy left hyperideal of , by Remark 15, we have and . Thus, we have
This implies that .

(6)(1) Assume that (6) holds. Let and be any -fuzzy right hyperideal and any -fuzzy left hyperideal of , respectively. Since is an -fuzzy bi-hyperideal of , by the assumption and Lemma 9, we have
Hence, . Therefore, is hyperregular by Theorem 23.

Similarly, we can show that (1)(7). This completes the proof.

Corollary 32. *Let be a semihyperring. Then the following conditions are equivalent.*(1)* is hyperregular.*(2)* for every bi-hyperideal and every left hyperideal of .*(3)* for every quasi-hyperideal and every left hyperideal of .*(4)* for every right hyperideal and every bi-hyperideal of .*(5)* for every right hyperideal and every quasi-hyperideal of .*(6)* for every right hyperideal , every bi-hyperideal , and every left hyperideal of .*(7)* for every right hyperideal , every quasi-hyperideal , and every left hyperideal of .*

*Definition 33. *A subset in a semihyperring is called idempotent if . A fuzzy subset in a semihyperring is called -idempotent if .

*Example 34. *Consider Example 21. Let . Then and so is idempotent. Define a fuzzy subset of by , and . Then and so is -fuzzy idempotent.

Theorem 35. *A semihyperring is hyperregular if and only if all -fuzzy right and -fuzzy left hyperideals of are -fuzzy idempotent and for every -fuzzy right hyperideal and every -fuzzy left hyperideal of , the fuzzy subset is an -fuzzy quasi-hyperideal of . *

*Proof. *Assume that is hyperregular. Let and be any -fuzzy right hyperideal and any -fuzzy left hyperideal of , respectively. Then we have . Since is hyperregular, by Theorem 31, we have and so . Hence is -idempotent. In a similar way, we may prove that every -fuzzy left hyperideal is -idempotent. Now let and be any -fuzzy right hyperideal and any -fuzzy left hyperideal of , respectively. By Theorem 23, we have and so
Therefore is an -fuzzy quasi-hyperideal of .

Conversely, assume that the given conditions hold. Let be any -fuzzy quasi-hyperideal of . It is easy to check that is an -fuzzy left hyperideal of ; then by the assumption, it is -fuzzy idempotent. Thus, we have
That is, . Similarly, we can show that . Thus,
On the other hand, it is clear that and are an -fuzzy right hyperideal and an -fuzzy left hyperideal of , respectively. By the assumption, we have , , and is an -fuzzy quasi-hyperideal of . Thus, we have
which implies that . By Theorem 27, is hyperregular.

Corollary 36. *A semihyperring is hyperregular if and only if all right hyperideals and all left hyperideals of are idempotent and for every right hyperideal and every left hyperideal of *