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Journal of Mathematics
Volume 2013 (2013), Article ID 510790, 10 pages
http://dx.doi.org/10.1155/2013/510790
Research Article

On Intra-Regular Left Almost Semihypergroups with Pure Left Identity

1Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
2Department of Civil Engineering and Architecture, Via delle Scienze 206, Udine, Italy
3Department of Mathematics, COMSATS Institute of Information Technology, Attock, Pakistan

Received 24 January 2013; Accepted 4 April 2013

Academic Editor: Feng Feng

Copyright © 2013 Naveed Yaqoob 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

We characterize intra-regular LA-semihypergroups by using the properties of their left and right hyperideals, and we investigate some useful conditions for an LA-semihypergroup to become an intra-regular LA-semihypergroup.

1. Introduction

Kazim and Naseeruddin [1] introduced the concept of left almost semigroups (abbreviated as LA-semigroups) and right almost semigroups (abbreviated as RA-semigroups). They generalized some useful results of semigroup theory. Later, Mushtaq [2, 3] and others further investigated the structure and added many useful results to the theory of LA-semigroups; see also [49]. An LA-semigroup is the midway structure between a commutative semigroup and a groupoid. It nevertheless possesses many interesting properties which we usually find in commutative and associative algebraic structures. Mushtaq and Yusuf produced useful results [10] on locally associative LA-semigroups in 1979. In this structure, they defined powers of an element and congruences using these powers. There are several results which have been added to the theory of LA-semigroups by Mushtaq, Shabir, Aslam, Davvaz, Madad, Hila, Chinram, Holgate, Jezek, Protic, and many other researchers.

Hyperstructure theory was introduced in 1934, when Marty [11] defined hypergroups, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science, and they are studied in many countries of the world. 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. A lot of papers and several books have been written on hyperstructure theory; see [12, 13]. A recent book on hyperstructures [14] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Another book [15] is devoted especially to the study of hyperring theory. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: -hyperstructures and transposition hypergroups. Many authors studied different aspects of semihypergroups, for instance, Aslam et al. [1620], Corsini et al. [21, 22], Davvaz et al. [23, 24], Hasankhani [25], Hila et al. [26], Leoreanu [27], Onipchuk [28], and Yaqoob et al. [2934].

Recently, Hila and Dine [35] introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semigroups. They investigated several properties of hyperideals of LA-semihypergroup and defined the topological space and studied the topological structure of LA-semihypergroups using hyperideal theory.

In this paper, we will prove some results on intra-regular LA-semihypergroups.

2. Basic Definitions and Examples

In this section, we recall certain definitions and results needed for our purpose.

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.

If and are two nonempty subsets of , then we denote

Definition 2 (see [35]). A hypergroupoid is called an LA-semihypergroup if for all ,

The law is called a left invertive law.

Example 3. Let with the binary hyperoperation defined below: Clearly, is not a semihypergroup because . Thus, is an LA-semihypergroup because the elements of satisfy the left invertive law.

Example 4. Let . If we define , where . Then, becomes an LA-semihypergroup as This implies that holds for all , and also it is clear that . Hence is an LA-semihypergroup.

Every LA-semihypergroup satisfies the law for all . This law is known as medial law (cf. [35]).

Definition 5. Let be an LA-semihypergroup. An element is called(i)left identity (resp., pure left identity) if for all , (resp., ),(ii)right identity (resp., pure right identity) if for all , (resp., ),(iii)identity (resp., pure identity) if for all , (resp., ).

Example 6. Let with the binary hyperoperation defined below: Clearly is an LA-semihypergroup because the elements of satisfy the left invertive law. Here, is a pure left identity because for all , . And, in Example 6, one can see that is a left identity but not a pure left identity.

Lemma 7. Let be an LA-semihypergroup with pure left identity . Then, holds for all .

Proof. Let be an LA-semihypergroup with pure left identity . Then, for all and by medial law, we have This completes the proof.

Lemma 8. Let be an LA-semihypergroup with pure left identity . Then, holds for all .

Proof. Let be an LA-semihypergroup with pure left identity . Then, for all and by medial law, we have This completes the proof.

The law is called a paramedial law.

Definition 9. Let be an LA-semihypergroup. A nonempty subset of is called a sub LA-semihypergroup of if for every .

Definition 10. A subset of an LA-semihypergroup is called a right (left) hyperideal of if () and is called a hyperideal if it is two-sided hyperideal.

Definition 11. By a bi-hyperideal of an LA-semihypergroup , we mean a sub LA-semihypergroup of such that .

Definition 12. A sub LA-semihypergroup of is called a -hyperideal of if .

It is easy to note that each right hyperideal is a bi-hyperideal. If denotes the set of all idempotent subsets of with pure left identity , then forms a hypersemilattice structure. The intersection of any set of bi-hyperideals of an LA-semihypergroup is either empty or a bi-hyperideal of .

Definition 13. A sub LA-semihypergroup of is called an interior hyperideal of if .

Definition 14. A nonempty subset of an LA-semihypergroup is called a quasi-hyperideal of if .

Lemma 15. If is an LA-semihypergroup with left identity , then .

Proof. If is an LA-semihypergroup with left identity , then implies that That is, .

Corollary 16. If is an LA-semihypergroup with pure left identity , then and .

Proof. The proof is similar to the proof of Lemma 15.

In an LA-semigroup, every right identity becomes a left identity. But in an LA-semihypergroup, every right identity needs not to be a left identity. For this, let with the binary hyperoperation defined below: Clearly, is not a semihypergroup because . Thus, is an LA-semihypergroup because the elements of satisfy the left invertive law. Here, is a right identity but not a left identity. However, if an LA-semihypergroup has a pure right identity , then becomes a pure left identity. For this, consider and be a pure right identity of then; This shows that in an LA-semihypergroup, every pure right identity becomes a pure left identity. Every LA-semihypergroup with pure right identity becomes a commutative hypermonoid.

Theorem 17. An LA-semihypergroup is a semihypergroup if and only if holds for all .

Proof. Let be a semihypergroup. Then, we have for all , but ; thus, On the other hand, suppose that holds for all . Since is an LA-semihypergroup, therefore Thus, is a semihypergroup. This completes the proof.

3. Intra-Regular LA-Semihypergroups

In this section, we will characterize intra-regular LA-semihypergroup by using the properties of left and right hyperideals.

Definition 18. Let be an LA-semihypergroup and . Then, is said to be regular if there exist an element such that . The LA-semihypergroup is called regular if every element of is regular.

Definition 19. An element of an LA-semihypergroup is called an intra-regular element if there exist such that and is called intra-regular, if every element of is intra-regular.

Example 20. Let with the binary hyperoperation defined below: Clearly, is an LA-semihypergroup because the elements of satisfy the left invertive law. Here, is intra-regular because, , , , .

Example 21. Let . Define a hyperoperation on by Then, for all , we have This implies that is a an LA-semihypergroup. Since and also . Thus, is a regular as well as intra-regular LA-semihypergroup.

Definition 22. An element of an LA-semihypergroup with left identity is called a left (right) invertible if there exist such that () and is called invertible if it is both a left and a right invertible. An LA-semihypergroup is called a left (right) invertible if every element of is a left (right) invertible and is called invertible if it is both a left and a right invertible.

Theorem 23. Every LA-semihypergroup with pure left identity is intra-regular if is left (right) invertible.

Proof. Let be a left invertible LA-semihypergroup with pure left identity . Then, for , there exist such that . Now, by using left invertive law, and medial law, Lemmas 7 and 15, we have This shows that is intra-regular. The case for right invertible can be seen in a similar way.

Theorem 24. An LA-semihypergroup with left identity is intra-regular if or for all .

Proof. Let be an LA-semihypergroup such that holds for all . Then, . Let , therefore, by using medial law, we have This shows that is intra-regular.
Let and assume that holds for all . Then, by using left invertive law, we have Thus, holds for all ; therefore, it follows from above that is intra-regular.

Corollary 25. An LA-semihypergroup with pure left identity is intra-regular if or for all .

Corollary 26. If is an LA-semihypergroup such that holds for all , then holds for all .

Theorem 27. If is an intra-regular LA-semihypergroup with pure left identity , then , where is a bi- (generalized bi-) hyperideal of .

Proof. Let be an intra-regular LA-semihypergroup with pure left identity . Then, clearly, . Now let . Since is intra-regular, so there exist such that . Now, by using Lemma 7, left invertive law, paramedial law and medial law, we have This shows that .

The converse is not true in general. For this, let us consider an LA-semihypergroup with the binary hyperoperation defined below: Clearly, is an LA-semihypergroup because the elements of satisfy the left invertive law. It is easy to see that is a bi-hyperideal of such that , but has no pure left identity, and also is not an intra-regular because is not an intra-regular.

Theorem 28. If is an intra-regular LA-semihypergroup with pure left identity , then , where is an interior hyperideal of .

Proof. Let be an intra-regular LA-semihypergroup with pure left identity . Then, clearly, . Now, let . Since is an intra-regular, so there exist such that . Now, by using paramedial law and left invertive law, we have This shows that .

The converse is not true in general. For this, let us consider an LA-semihypergroup with the binary hyperoperation defined below: Clearly, is an LA-semihypergroup because the elements of satisfy the left invertive law. It is easy to see that is an interior hyperideal of with pure left identity such that , but is not an intra-regular because , is not an intra-regular.

Definition 29. Let be an LA-semihypergroup. Then, is called semiprime if implies .

Theorem 30. An LA-semihypergroup with pure left identity is intra-regular if , where and are the left and right hyperideals of , respectively, such that is semiprime.

Proof. Let be an LA-semihypergroup with pure left identity. Then, clearly, and are the left and right hyperideals of such that and , because by using paramedial law, we have Therefore, by the given assumption, . Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have This shows that is intra-regular.

Lemma 31. If is an intra-regular LA-semihypergroup, then .

Proof. The proof is straightforward.

Theorem 32. For a left invertible LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is intra-regular,(ii), where and are any right and left hyperideals of , respectively.

Proof. : assume that is an intra-regular LA-semihypergroupwith pure left identity and let . Then, there exist such that . Let and be any right and left hyperideals of , respectively. Then, obviously, . Now, let imply that and . Now by using medial law, left invertive law, and Lemma 7, we have This shows that .
let be a left invertible LA-semihypergroup with pure left identity. Then, for . There, exists such that . Since is a right hyperideal and also a left hyperideal of such that , therefore, by using the given assumption, medial law, left invertive law, and Lemma 7, we have Thus, we get for some .
Now, by using left invertive law, we have This shows that is intra-regular.

Lemma 33. Every left hyperideal of an intra-regular LA-semihypergroup with pure left identity is idempotent.

Proof. The proof is straightforward.

Theorem 34. In an LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is intra-regular,(ii), where is any left hyperideal of .

Proof. : let be a left hyperideal of an intra-regular LA-semihypergroup with left identity. Then, and, by Lemma 33, . Now , which implies that .
: let be a left hyperideal of . Then, , which implies that is idempotent; hence, is intra-regular.

Theorem 35. In an intra-regular LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is a bi- (generalized bi-) hyperideal of ,(ii) and .

Proof. : let be a bi-hyperideal of an intra-regular LA-semihypergroup with pure left identity. Then, . Let . Then, since is intra-regular, so there exist , such that . Now, by using medial, left invertive law, and Lemma 7, we have Thus, holds. Now, by using left invertive law, paramedial law, medial law, and Lemma 7, we have Hence, holds.
is obvious.

Theorem 36. In an intra-regular LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is a quasi-hyperideal of ,(ii).

Proof. : let be a quasi hyperideal of an intra-regular LA-semihypergroup with pure left identity. Then, . Let , then since is intra-regular, so there exist such that . Let . Then, by using medial law, paramedial law and Lemma 7, we have Now, let . Then, by using medial law, paramedial law and Lemma 7, we have Hence, . As, by using left invertive law, medial law, and Lemma 7, we have Thus, implies that .
is obvious.

Theorem 37. In an intra-regular LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is an interior hyperideal of ,(ii).

Proof. : let be an interior hyperideal of an intra-regular LA-semihypergroup with pure left identity. Then, . Let , then since is intra-regular, so there exist such that . Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have Thus, .
is obvious.

Theorem 38. In an intra-regular LA-semihypergroup with pure left identity, the following conditions are equivalent.(i) is a -hyperideal of ,(ii) and .

Proof. : let be a -hyperideal of an intra-regular LA-semihypergroup with pure left identity. Then, and . Let . Then, since is intra-regular, so there exist such that . Now, by using left invertive law, medial law, paramedial law and Lemma 7, we have Thus, . Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we haveHence, .
is obvious.

Lemma 39. Every nonempty subset of an intra-regular LA-semihypergroup with pure left identity is a left hyperideal of if and only if it is a right hyperideal of .

Proof. The proof is straightforward.

Theorem 40. In an intra-regular LA-semihypergroup with pure left identity, the following conditions are equivalent:(i) is a -hyperideal of ,(ii) is a two-sided hyperideal of .

Proof. : assume that is intra-regular LA-semihypergroup with pure left identity, and let be a -hyperideal of . Then, . Let . Then, since is intra-regular, so there exist such that . Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have Hence, is a left hyperideal of , and, by Lemma 39, is a two-sided hyperideal of .
: let be a two-sided hyperideal of . Let . Then, for some . Now, by using Lemma 7, we have Hence, ; therefore, is a -hyperideal of .

Lemma 41. Every nonempty subset of an intra-regular LA-semihypergroup with pure left identity is a two-sided hyperideal of if and only if it is a quasi hyperideal of .

Proof. The proof is straightforward.

Theorem 42. A two-sided hyperideal of an intra-regular LA-semihypergroup with pure left identity is minimal if and only if it is the intersection of two minimal two-sided hyperideals of .

Proof. Let be an intra-regular LA-semihypergroup and a minimal two-sided hyperideal of . Let . As and , which shows that and are left hyperideals of . So, by Lemma 39, and are two-sided hyperideals of . Now, This implies that is a quasi hyperideal of , so by using Lemma 41, is a two-sided hyperideal of . Also since , we have Now, since is minimal, so , where and are minimal two-sided hyperideals of . Let, be any two-sided hyperideal of such that . Then, , which implies that . Thus . Therefore, we have Thus, is a minimal two-sided hyperideal of . Similarly, is a minimal two-sided hyperideal of .
Conversely, let be a two-sided hyperideal of , where and are minimal two-sided hyperideals of . Then, by using Lemma 41, is a quasi hyperideal of ; that is, . Let be a two-sided hyperideal of such that . Then, Now which implies that is a left hyperideal and, hence, a two-sided hyperideal by Lemma 39. Similarly, is a two-sided hyperideal of .
But since and are minimal two-sided hyperideals of , therefore, and . But , which implies that . This gives us that and, hence, is minimal.

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