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Volume 2011 (2011), Article ID 953124, 8 pages
On Hyperideals in Left Almost Semihypergroups
Department of Mathematics & Computer Science, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra 6001, Albania
Received 7 June 2011; Accepted 11 July 2011
Academic Editors: A. V. Kelarev and A. Kiliçman
Copyright © 2011 Kostaq Hila and Jani Dine. 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.
This paper deals with a class of algebraic hyperstructures called left almost semihypergroups (LA-semihypergroups), which are a generalization of LA-semigroups and semihypergroups. We introduce the notion of LA-semihypergroup, the related notions of hyperideal, bi-hyperideal, and some properties of them are investigated. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks, and so forth. We define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.
1. Introduction and Preliminaries
The applications of mathematics in other disciplines, for example in informatics, play a key role, and they represent, in the last decades, one of the purposes of the study of the experts of hyperstructures theory all over the world. Hyperstructure theory was introduced in 1934 by a French mathematician Marty , at the 8th Congress of Scandinavian Mathematicians, where he defined hypergroups based on the notion of hyperoperation, 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 and for their applications to many subjects of pure and applied mathematics and computer science by many mathematicians. 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. Some principal notions about hyperstructures and semihypergroups theory can be found in [1–7].
The Theory of ideals, in its modern form, is a contemporary development of mathematical knowledge to which mathematicians of today may justly point with pride. Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such as in informatics, physics, and others. As an example of applications of the concept of an ideal in informatics, let us mention that ideals of algebraic structures have been used recently to design efficient classification systems, see [8–12].
The study of LA-semigroup as a generalization of commutative semigroup was initiated in 1972 by Kazim and Naseeruddin . They have introduced the concept of an LA-semigroup and have investigated some basic but important characteristics of this structure. They have generalized some useful results of semigroup theory. Since then, many papers on LA-semigroups appeared showing the importance of the concept and its applications [13–23]. In this paper, we generalize this notion introducing the notion of LA-semihypergroup which is a generalization of LA-semigroup and semihypergroup, proposing so a new kind of hyperstructure for further studying. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks etc. Although the hyperstructure is nonassociative and noncommutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic hyperstructures. A several properties of hyperideals of LA-semihypergroup are investigated. In this note, we define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.
Recall first the basic terms and definitions from the hyperstructure theory.
Definition 1.1. 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 .
Definition 1.2. A hyperstructure is called the pair , where is a hyperoperation on the set .
Definition 1.3. A hyperstructure is called a semihypergroup if for all , , which means that
If and are nonempty subsets of , then
Definition 1.4. A nonempty subset of a semihypergroup is called a sub-semihypergroup of if , and is called in this case super-semihypergroup of .
Definition 1.5. Let be a semihypergroup. Then is called a hypergroup if it satisfies the reproduction axiom, for all , .
Definition 1.6. A hypergrupoid is called an LA-semihypergroup if, for all ,
Every LA-semihypergroup satisfies the medial law, that is, for all ,
In every LA-semihypergroup with left identity, the following law holds: for all .
An element in an LA-semihypergroup is called identity if . An element 0 in a semihypergroup is called zero element if . A subset of an LA-semihypergroup is called a right (left) hyperideal if and is called a hyperideal if it is two-sided hyperideal, and if is a left hyperideal of , then becomes a hyperideal of . By a bi-hyperideal of an LA-semihypergroup , we mean a sub-LA-semihypergroup of such that . It is easy to note that each right hyperideal is a bi-hyperideal. If has a left identity, then it is not hard to show that is a bi-hyperideal of and . If denotes the set of all idempotents subsets of with left identity , then forms a hypersemilattice structure, also if , then . The intersection of any set of bi-hyperideals of an LA-semihypergroup is either empty or a bi-hyperideal of . Also the intersection of prime bi-hyperideals of an LA-semihypergroup is a semiprime bi-hyperideal of .
2. Main Results
Proposition 2.1. Let be an LA-semihypergroup with left identity, a left hyperideal, and a bi-hyperideal of . Then and are bi-hyperideals of .
Proof. Using the medial law (1.4), we get also Hence, is a bi-hyperideal of . we obtain also Hence, is a bi-hyperideal of .
Proposition 2.2. Let be an LA-semihypergroup with left identity and two bi-hyperideals of . Then is a bi-hyperideal of .
Proof. Using (1.4), we get
By the above, if and are nonempty, then and are connected bi-hyperideals. Proposition 2.1 leads us to an easy generalization, that is, if are bi-hyperideals of an LA-semihypergroup with left identity, then are bi-hyperideals of , consequently the set of bi-hyperideals forms an LA-semihypergroup.
If is an LA-semihypergroup with left identity , then and are bi-hyperideals of . It can be easily shown that , , and . Hence, this implies that and . Also, , , , , and (if is an idempotent), consequently . It is easy to show that .
Lemma 2.3. Let be an LA-semihypergroup with left identity, and let be an idempotent bi-hyperideal of . Then is a hyperideal of .
Proof. By the definition of LA-semihypergroup (1.3), we have and every right hyperideal in with left identity is left.
Lemma 2.4. Let be an LA-semihypergroup with left identity , and let be a proper bi-hyperideal of . Then .
Proof. Let us suppose that . Since , using (1.3), we have . It is impossible. So, .
It can be easily noted that .
Proposition 2.5. Let be an LA-semihypergroup with left identity, and let be bi-hyperideals of . Then the following statements are equivalent: (1)every bi-hyperideal is idempotent, (2),(3)the hyperideals of form a hypersemilattice , where .
Proof. (1)⇒(2). Using Lemma 2.3, it is easy to note that . Since implies , hence .
(2)⇒(3). and . Similarly, associativity follows. Hence, is a hypersemilattice.
A bi-hyperideal of an LA-semihypergroup is called a prime bi-hyperideal if implies either or for every bi-hyperideal and of . The set of bi-hyperideals of is totally ordered under the set inclusion if for all bi-hyperideals either or .
Theorem 2.6. Let be an LA-semihypergroup with left identity. Every bi-hyperideal of is prime if and only if it is idempotent and the set of the bi-hyperideals of is totally ordered under the set inclusion.
Proof. Let us assume that every bi-hyperideal of is prime. Since is a hyperideal and so is prime which implies that , hence is idempotent. Since is a bi-hyperideal of (where and are bi-hyperideals of ) and so is prime. Now by Lemma 2.3, either or which further implies that either or . Hence, the set of bi-hyperideals of is totally ordered under set inclusion.
Conversely, let us assume that every bi-hyperideals of is idempotent and the set of bi-hyperideals of is totally ordered under set inclusion. Let and be the bi-hyperideals of with and without loss of generality assume that . Since is an idempotent, so implies that , and, hence, every bi-hyperideal of is prime.
A bi-hyperideal of an LA-semihypergroup is called strongly irreducible bi-hyperideal if implies either or for every bi-hyperideal and of .
Theorem 2.7. Let be an LA-semihypergroup with zero. Let be the set of all bi-hyperideals of , and the set of all strongly irreducible proper bi-hyperideals of , then forms a topology on the set , where and : Bi-hyperideal preserves finite intersection and arbitrary union between the set of bi-hyperideals of and open subsets of .
Proof. Since is a bi-hyperideal of and 0 belongs to every bi-hyperideal of , then , also which is the first axiom for the topology. Let , then , where is a bi-hyperideal of generated by . Let and , if , then and . Let us suppose , this implies that either or . It is impossible. Hence, which further implies that . Thus . Now if , then and . Thus and , therefore , which implies that . Hence is the topology on . Define : Bi-hyperideal by , then it is easy to note that preserves finite intersection and arbitrary union.
A hyperideal of an LA-semihypergroup is called prime if implies that either or for all hyperideals and in .
Let denotes the set of proper prime hyperideals of an LA-semihypergroup absorbing 0. For a hyperideal of , we define the sets and .
Theorem 2.8. Let be an LA-semihypergroup with zero. The set constitutes a topology on the set .
Proof. Let , if , then and and . Let which implies that either or , which is impossible. Hence, . Similarly . The remaining proof follows from Theorem 2.7.
The assignment preserves finite intersection and arbitrary union between the hyperideal and their corresponding open subsets of .
Let be a left hyperideal of an LA-semihypergroup . is called quasiprime if for left hyperideals of such that , we have or .
Theorem 2.9. Let be an LA-semihypergroup with left identity . Then a left hyperideal of is quasiprime if and only if implies that either or .
Proof. Let be a left hyperideal of . Let us assume that , then
Hence, either or .
Conversely, let us assume that , where and are left hyperideal of such that . Then there exists such that . Now, by the hypothesis, we have for all . Since , so by hypothesis, for all , we obtain . This shows that is quasiprime.
An LA-semihypergroup is called an antirectangular if , for all . It is easy to see that . In the following results for an antirectangular LA-semihypergroup , .
Proposition 2.10. Let be an LA-semihypergroup. If are hyperideals of , then is a hyperideal.
Proof. Using (1.4), we have also which shows that is a hyperideal.
Consequently, if are hyperideals of , then are hyperideals of and the set of hyperideals of form an antirectangular LA-semihypergroup.
Lemma 2.11. Let be an antirectangular LA-semihypergroup. Any subset of is left hyperideal if and only if it is right.
It is fact that . From the above lemma, we remark that every quasiprime hyperideal becomes prime in an antirectangular LA-semihypergroup.
Lemma 2.12. Let be an anti-rectangular LA-semihypergroup. If is a hyperideal of , then .
Proof. Let , then . Hence . Also, .
An hyperideal of an LA-semihypergroup is called an idempotent if . An LA-semihypergroup is said to be fully idempotent if every hyperideal of is idempotent.
Proposition 2.13. Let be an antirectangular LA-semihypergroup, and, be hyperideals of . Then the following statements are equivalent: (1) is fully idempotent, (2),(3)the hyperideals of form a hypersemilattice where .
The proof follows from Proposition 2.5.
The set of hyperideals of is totally ordered under set inclusion if for all hyperideals either or and denoted by hyperideal.
Theorem 2.14. Let be an antirectangular LA-semihypergroup. Then every hyperideal of is prime if and only if it is idempotent and hyperideal is totally ordered under set inclusion.
Proof. The proof follows from Theorem 2.6.
In conclusion, let us mention that it would be interesting to investigate whether it is possible to apply hyperideals of hyperstructures to the construction of classification systems similar to those introduced in [8–12].
The authors are highly grateful to referees for their valuable comments and suggestions.
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