On Hyperideals in Left Almost Semihypergroups

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.


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 1 , 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 2 ISRN Algebra 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 13 . 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 LAsemihypergroup 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.
In every LA-semihypergroup with left identity, the following law holds:

Main Results
Hence, T 2 • B is a bi-hyperideal of H. Proof. Using 1.4 , we get

2.5
By the above, if B 1 and B 2 are nonempty, then B 1 • B 2 and B 2 • B 1 are connected bihyperideals. Proposition 2.1 leads us to an easy generalization, that is, if B 1 , B 2 , B 3 , . . . , B n are bi-hyperideals of an LA-semihypergroup H with left identity, then   B 1 ⊆ B or B 2 ⊆ B for every bi-hyperideal B 1 and B 2  Proof. Let us assume that every bi-hyperideal of H is prime. Since B 2 is a hyperideal and so is prime which implies that B ⊆ B • B, hence B is idempotent. Since B 1 ∩ B 2 is a bi-hyperideal of H where B 1 and B 2 are bi-hyperideals of H and so is prime. Now by Lemma 2.3, either B 1 ⊆ B 1 ∩ B 2 or B 2 ⊆ B 1 ∩ B 2 which further implies that either B 1 ⊆ B 2 or B 2 ⊆ B 1 . Hence, the set of bi-hyperideals of H is totally ordered under set inclusion.
Conversely, let us assume that every bi-hyperideals of H is idempotent and the set of bi-hyperideals of H is totally ordered under set inclusion. Let B 1 , B 2 and B be the bihyperideals of H with B 1 • B 2 ⊆ B and without loss of generality assume that  Proof. Let Θ I 1 , Θ I 2 ∈ Γ P H , if J ∈ Θ I 1 ∩ Θ I 2 , then J ∈ P H and I 1 ⊆J and I 2 ⊆J. Let I 1 ∩ I 2 ⊆ J which implies that either I 1 ⊆ J or I 2 ⊆ J, which is impossible. Hence, J ∈ Θ I 1 ∩I 2 . Similarly Θ I 1 ∩I 2 ⊆ Θ I 1 ∩ Θ I 2 . The remaining proof follows from Theorem 2.7.
The assignment I → Θ I preserves finite intersection and arbitrary union between the hyperideal H and their corresponding open subsets of Θ I .
Let P be a left hyperideal of an LA-semihypergroup H. P is called quasiprime if for left hyperideals A, B of H such that A • B ⊆ P , we have A ⊆ P or B ⊆ P .