Journal of Applied Mathematics

Volume 2014 (2014), Article ID 658252, 6 pages

http://dx.doi.org/10.1155/2014/658252

Research Article

Generalized Rough Γ-Hyperideals in Γ-Semihypergroups

Naveed Yaqoob1,2 and Shamsul Haq³

¹Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

²Department of Mathematics, College of Science in Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia

³Natural Resource and Engineering Department, University of Kurdistan Hewlr, Kurdistan Region, Iraq

Received 1 May 2014; Accepted 17 June 2014; Published 8 July 2014

Academic Editor: Ashraf Zenkour

Copyright © 2014 Naveed Yaqoob and Shamsul Haq. 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.

Davvaz (2008) introduced the concept of set-valued homomorphism and -rough sets in a group. In this paper, we consider the set-valued homomorphism on -semihypergroup to interpret the lower and upper approximations. We study the roughness of bi--hyperideals and quasi--hyperideals in terms of set-valued homomorphisms, which are extended notions of bi--hyperideals and quasi--hyperideals of -semihypergroups.

1. Introduction

Hyperstructure, in particular hypergroups, was introduced in 1934, by Marty [1]. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science and they are studied in many countries in the world. Recently, Anvariyeh et al. [2] introduced the notion of -semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup, and a generalization of a -semigroup. Heidari et al. [3] studied the structure further and added some useful results to the theory of -semihypergroups. Abdullah et al. [4–6] studied some properties of -hypersystems and bi--hyperideals in -semihypergroups; also Hila et al. [7] studied the structures of -semihypergroups.

In 1982, Pawlak [8] introduced the notion of rough sets as a tool to model uncertainty and vague and incomplete information system. The theory of rough sets is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. At present, this concept has been applied to many directions, such as groups, probability theory, graph theory, automata theory, topology, cognitive sciences, machine learning, knowledge acquisition, and pattern recognition. The algebraic approach to rough sets has been studied by some authors; for instance, Biswas and Nanda [9] studied the classical group theory in terms of rough sets and introduced the notion of rough subgroups. Xiao and Zhang [10] introduced the notion of rough prime ideals and rough fuzzy prime ideals in a semigroup. Kuroki [11] introduced the notion of a rough ideal in a semigroup. In [12] the authors investigated Pawlak’s approximations in -semihypergroups. Aslam et al. [13] introduced the concept of rough -hypersystems and fuzzy -hypersystems in -semihypergroups. Yaqoob et al. [14–16] applied rough set theory to bi--hyperideals and quasi--hyperideals in -semihypergroups. Fotea [17] and Leoreanu-Fotea and Davvaz [18] discussed the lower and upper approximations of hypergroups and n-ary hypergroups. Jun [19] studied the roughness of -subsemigroups/ideals in -semigroups.

In 2008, Davvaz [20] introduced the concept of set-valued homomorphism and -rough sets in a group. This idea attracted several mathematicians. Xiao [21] studied the properties of -roughness in semigroups. Yamak et al. [22] introduced generalized lower and upper approximations in a ring. Yaqoob and Aslam [23] applied generalized rough set theory (in terms of set-valued homomorphisms) to the theory of -semihypergroups. Hosseini et al. [24] applied -rough set theory to semigroups.

In this paper, we study the roughness of bi--hyperideals and quasi--hyperideals in terms of set-valued homomorphisms.

2. Preliminaries and Basic Definitions

In this section, we will recall some concepts related to -semihypergroups and generalized rough sets. Throughout the paper, denotes a -semihypergroup unless otherwise specified.

Definition 1. A map is called hyperoperation or join operation on the set , where is a nonempty set and denotes the set of all nonempty subsets of . A hypergroupoid is a set together with a (binary) hyperoperation. A hypergroupoid , which is associative, that is, , for all , is called a semihypergroup.

Definition 2 (see [2]). Let and be two nonempty sets. is called a -semihypergroup if every is a hyperoperation on , that is, , for every , and, for every and , we have .

Let and be two nonempty subsets of . Then, we define

Let be a semihypergroup and let . Then, is a -semihypergroup. So, every semihypergroup is -semihypergroup.

Definition 3 (see [2]). Let be a -semihypergroup and . A nonempty subset of is called a sub--semihypergroup of if for every .

Definition 4 (see [2]). A subset of a -semihypergroup is called an interior -hyperideal of if .

Let be a nonempty subset of and . A set is defined to be the set

For example, and .

Definition 5 (see [14]). A subset of a -semihypergroup is called an   -hyperideal (resp.,   -hyperideal) of if (resp., ).

Definition 6 (see [14]). A sub--semihypergroup of a -semihypergroup is called an bi--hyperideal of , if , where , are nonnegative integers is suppressed if .

Definition 7 (see [15]). An bi--hyperideal of a -semihypergroup is called prime if for ,     or , for all implies or , for all .

Definition 8 (see [15]). An bi--hyperideal of a -semihypergroup is called semiprime if for  ,    or , for all implies , for all .

Definition 9 (see [6]). Let be a -semihypergroup and a sub--semihypergroup of . Then is called an -left -hyperideal of if where is any positive integer. Dually, , and then is called an -right -hyperideal of , where is any positive integer.

Definition 10 (see [6]). Let be a -semihypergroup and a nonempty subset of . Then is called an quasi--hyperideal of if .

Now, we will recall some notions in generalized rough sets.

Definition 11. Let and be two nonempty universes. Let be a set-valued mapping given by . Then, the triple is referred to as a generalized approximation space or generalized rough set. Any set-valued function from to defines a binary relation from to by setting . Obviously, if is an arbitrary relation from to , then it can be defined as a set-valued mapping by where . For any set , the lower and upper approximations and are defined by

The pair is referred to as a generalized rough set, and , are referred to as lower and upper generalized approximation operators, respectively.

If a subset satisfies that , then is called a definable set of . We denote all the definable sets of by Def .

Theorem 12 (see [22]). Let be a generalized approximation space; its lower and upper approximation operators satisfy the following properties: for all ,

where is the complement of the set .

Theorem 13 (see [22]). Let be a generalized approximation space; its lower and upper generalized approximation operators satisfy the following properties: for all ,

If is an equivalence relation on , then the pair is the Pawlak approximation space. Therefore, a generalized rough set is an extended notion of Pawlak’s rough sets.

Definition 14. Let be a -semihypergroup. An equivalence relation on is called regular on if

for all and .

If is a regular relation on , then, for every , stands for the class of with the representation . A regular relation on is called complete if for all and . In addition, on is called congruence if, for every and , we have . It is obvious that, for a regular relation on ,   for all and .

3. Generalized Rough Sets in -Semihypergroups

In this section, we will present some results on generalized rough sets in -semihypergroups.

Definition 15 (see [23]). A set-valued homomorphism from a -semihypergroup to a -semihypergroup is a mapping from to which preserves the operation; that is, for all ,   , and .   is called a strong set-valued homomorphism, if for all ,  , and .

Theorem 16 (see [23]). Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism. If , are two nonempty subsets of , then

(1)  ;

(2) if is strong, then .

Theorem 17 (see [23]). Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism.

(1) If is a sub--semihypergroup of , then is, if it is nonempty, a sub--semihypergroup of .

(2) If is a left (resp., right) -hyperideal of , then is, if it is nonempty, a left (resp., right) -hyperideal of .

Theorem 18 (see [23]). Let be a -semihypergroup, let be a -semihypergroup, and let be a strong set-valued homomorphism.

(1) If is a sub--semihypergroup of , then is, if it is nonempty, a sub--semihypergroup of .

(2) If A is a left (resp., right) -hyperideal of , then is, if it is nonempty, a left (resp., right) -hyperideal of .

Theorem 19. Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism. If is an interior -hyperideal of , then

(1) is, if it is nonempty, an interior -hyperideal of ;

(2) if is strong, then is, if it is nonempty, an interior -hyperideal of .

Proof. The proof is straightforward.

Lemma 20. Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism. Then, for a nonempty subset of ,

(1) for all ;

(2) if is strong, then for all .

Proof. The proof is straightforward.

4. Generalized Rough (Bi-)Quasi--Hyperideals

We will study here some properties of generalized lower and upper approximations of bi--hyperideals in -semihypergroup.

A subset of a -semihypergroup is called a generalized upper (resp., generalized lower) rough bi--hyperideal of if (resp., ) is an bi--hyperideal of .

Theorem 21. Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism. If is an bi--hyperideal of , then

(1) is, if it is nonempty, an bi--hyperideal of ;

(2) if is strong, then is, if it is nonempty, an bi--hyperideal of .

Proof. (1) Let be an bi--hyperideal of . Then, by Theorem 16 and Lemma 20, we have

From this and Theorem 17, we obtain that is an bi--hyperideal of .

(2) Let be an bi--hyperideal of . Then, by Theorem 16 and Lemma 20, we have

From this and Theorem 18, we obtain that is, if it is nonempty, an bi--hyperideal of . This completes the proof.

Corollary 22. Let be a -semihypergroup, let be a -semihypergroup, and let be a set-valued homomorphism. If is an   -hyperideal (resp.,   -hyperideal, -left -hyperideal, and -right -hyperideal) of , then

(1) is, if it is nonempty, an   -hyperideal (resp.,   -hyperideal, -left -hyperideal, and -right -hyperideal) of ;

(2) if   is strong, then is, if it is nonempty, an   -hyperideal (resp.,   -hyperideal, -left -hyperideal, and -right -hyperideal) of .

Proof. The proof is straightforward.

A subset of a -semihypergroup is called a generalized upper (resp., generalized lower) rough prime bi--hyperideal of if (resp., ) is a prime bi--hyperideal of .

Theorem 23. Let be a -semihypergroup, let be a -semihypergroup, and let be a strong set-valued homomorphism. If is a prime bi--hyperideal of , then

(1) is, if it is nonempty, a prime bi--hyperideal of ;

(2) is, if it is nonempty, a prime bi--hyperideal of .

Proof. Since is an bi--hyperideal of . by Theorem 21, we know that and are bi--hyperideals of .

(1) Let be any element of . Let and such that . Thus,

where . Thus, there exist ,  , and such that . Since is a prime bi--hyperideal, we have or . Now,

Thus, or . So or . Therefore, is a prime bi--hyperideal of .

(2) We suppose that is not a prime bi--hyperideal; then for there exist and any element , such that , but and . Thus and . Then, there exist

Now, for ,  , and , we have

This implies that . Since is a prime bi--hyperideal of , we have or . It contradicts the supposition. This means that is, if it is nonempty, a prime bi--hyperideal of .

The following example shows that the converse of Theorem 23 does not hold.

Example 24. Let and be the sets of binary hyperoperations defined as follows:

Clearly is a -semihypergroup. Let and be the sets of binary hyperoperations defined as follows:

Clearly is a -semihypergroup. Assume that ,  , and . Here, is a strong set-valued homomorphism from to . Now for ,   and . It is clear that and are prime bi--hyperideals of . But is not a sub--semihypergroup of ; hence, is not a prime bi--hyperideal of .

A subset of a -semihypergroup is called a generalized upper (resp., generalized lower) rough quasi--hyperideal of if (resp., ) is an quasi--hyperideal of .

Theorem 25. Let  be a -semihypergroup, let  be a -semihypergroup, and let be a strong set-valued homomorphism. If is an quasi--hyperideal of , then is, if it is nonempty, an quasi--hyperideal of .

Proof. Let be an quasi--hyperideal of ; that is, . Note that . Then, by Theorem 12(L3), Theorem 16(2), and Lemma 20(2), we have

This shows that is an quasi--hyperideal of .

The next theorem shows that the intersection of a generalized lower rough -left -hyperideal and a generalized lower rough -right -hyperideal of a -semihypergroup is a generalized lower rough quasi--hyperideal of .

Theorem 26. Let be a -semihypergroup, let be a -semihypergroup, and let be a strong set-valued homomorphism. Let and be a generalized lower rough -left -hyperideal and a generalized lower rough -right -hyperideal of , respectively. Then is, if it is nonempty, an quasi--hyperideal of .

Proof. Let and be a generalized lower rough -left -hyperideal and a generalized lower rough -right -hyperideal of , respectively. Then,

Now, we have

Hence, this shows that is a generalized lower rough quasi--hyperideal of .

Conflict of Interests

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

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