On Some Hyperideals in Ordered Semihypergroups

Categories and General Algebraic Structures with Application

In this paper, first we introduce the notions of an (m, n)hyperideal and a generalized (m, n)-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. Thereafter, we characterize (m, n)-regularity, (m, 0)-regularity, and (0, n)-regularity of an ordered semihypergroup in terms of its (m, n)-hyperideals, (m, 0)-hyperideals and (0, n)-hyperideals, respectively. The relations mI , In, H n m , and B n m on an ordered semihypergroup are, then, introduced. We prove that B n m ⊆ H n m on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the (m, 0)-regularity [(0, n)regularity] of an element induce the (m, 0)-regularity [(0, n)-regularity] of the whole H n m-class containing that element as well as the fact that (m, n)regularity and (m, n)-right weakly regularity of an element induce the (m, n)regularity and (m, n)-right weakly regularity of the whole B n m-class and H n mclass containing that element, respectively.

International Journal of Research in Academic World

In this paper, we study relative ordered (m, n)-hyperideals in ordered semihypergroups. We also study relative (m, 0)-hyperideals and relative (0, n)-hyperideals as well as characterize regular ordered semihypergroups, and obtain some results based on these relative hyperideals. We prove that the intersection of all relative ordered (m, n)-hyperideals of S containing s is a relative ordered (m, n)-hyperideal of S containing s. Suppose that (S,•,≤) is an ordered semihypergroup, A ⊆ S and m,n are positive integers. We prove that if R(m,0) and L(0,n) be the set of all relative ordered (m,0)-hyperideals and the set of all relative ordered (0,n)-hyperideals of S, respectively. Then the following assertions are true: i) S is relative (m,0)-regular if and only if for all R ∈ R(m,0), R = (Rm •A]A. ii) S is relative (0,n)-regular if and only if for all L ∈ R(0,n), L = (A•Ln]A. Furthermore, suppose that (S, •, ≤) is an ordered semihypergroup and m, n are non-negative integers. Let A ⊆ S. Suppose that A(m,n) is the set of all relative ordered (m, n)-hyperideals of S. Then, we have the following: S is (m,n)-regular ⇐⇒ ∀A ∈ A(m,n),A = (Am •A•An]A.

2021

The aim of this article is to study ordered semihypergroups in the framework of (M,N)-int-soft bi-hyperideals. In this paper, we introduce the notion of (M,N)-int-soft bi-hyperideals of ordered semihypergroups. Some properties of (M,N)-int-soft bi-hyperideals in ordered semihypergroups are provided. We show that every int-soft bi-hyperideal is an (M,N)-int-soft bi-hyperideals of S over U but the converse is not true which is shown with help of an example. We characterize left (M,N) simple and completely regular ordered semihypergroups by means of (M,N)-int-soft bi-hyperideals.

Fundamental Journal of Mathematics and Applications

In this article, we deal with ordered generalized $\Gamma$-hyperideals in ordered $\Gamma$-semihypergroups. In particular, we study $(m, n)$-regular ordered $\Gamma$-semihypergroups in terms of ordered $(m, n)$-$\Gamma$-hyperideals. Moreover, we obtain some ideal theoretic results in ordered $\Gamma$-semihypergroups.

2015

In this work, we attempt to investigate the connection between various types of ideals (for examples (m;n)-ideal, bi-ideal, interior-ideal, quasi-ideal, prime-ideal and maximal-ideal) of an or- dered semigroup (S; � ; � ) and the corresponding, hyperideals of its EL-hyperstructure (S; � ) (if exists). Moreover, we construct the class of EL--semihypergroup, associated to a partially-ordered - semigroup.

2020

In this paper, we introduce the concept of unionsoft (briefly, uni-soft) bi-hyperideal of an ordered semihypergroup. The notions of prime (strongly prime, semiprime, irreducible, and strongly irreducible) uni-soft bi-hyperideals in ordered semihypergroups are introduced and related properties are investigated. Numerous examples of these notions are given. The relationship between prime and strongly prime, irreducible and strongly irreducible uni-soft bi-hyperideals are considered and characterizations of these concepts are established. Regular and intra-regular ordered semihypergroups are characterized in terms of these notions.

Abul Basar; , Shahnawaz Ali, Poonam Kumar Sharma, Bhavanari Satyanarayana; Mohammad Yahya Abbasi, Ikonion Journal of Mathematics, 1(2), 34-45, 2019

The main purpose of this paper is to investigate ordered-semihypergroups in the general terms of ordered-hyperideals. We introduce ordered (generalized) (m; n)-hyperideals in orderedsemihypergroups. Then, we characterize ordered-semihypergroup by ordered (generalized) (0; 2)-hyperideals, ordered (generalized) (1; 2)-hyperideals and ordered (generalized) 0-minimal (0; 2)-hyperideals. Furthermore, we investigate the notion of ordered (generalized) (0; 2)-bi-hyperideals, ordered 0-(0; 2) bisimple ordered-semihypergroups and ordered 0-minimal (generalized) (0; 2)-bi-hyperideals in ordered-semihyperoups. It is proved that an ordered-semihypergroup S with a zero 0 is 0-(0; 2)-bisimple if and only if it is left 0-simple.

Journal of Algebraic Hyperstructures and Logical Algebras, 1999

In this paper, we introduce the concept of unionsoft (briefly, uni-soft) bi-hyperideal of an ordered semihypergroup. The notions of prime (strongly prime, semiprime, irreducible, and strongly irreducible) uni-soft bi-hyperideals in ordered semihypergroups are introduced and related properties are investigated. Numerous examples of these notions are given. The relationship between prime and strongly prime, irreducible and strongly irreducible uni-soft bi-hyperideals are considered and characterizations of these concepts are established. Regular and intra-regular ordered semihypergroups are characterized in terms of these notions.

International Journal of Mathematical Analysis, 2013

In this paper, we introduce the concepts of bi-hyperideals, quasihyperideals, left hyperideals and right hyperideals in ordered Γ-semihypergroups and characterize intra-regular ordered Γ-semihypergroups in terms of their bi-hyperideals and quasi-hyperideals, bi-hyperideals and left hyperideals, bi-hyperideals and right hyperideals.

2019

A semigroup is an algebraic structure together with a nonempty set and an associative binary operation. The systematic study of semigroups started in the early 20th century. Semigroups are important in different areas of Mathematics. The concept of hyperstructures was introduced in 1934 as a suitable generalization of classical algebraic structures by Marty [1]. He obtained various results on hypergroups and applied them in different areas, for instance, in algebraic rational fractions, functions, and noncommutative groups. Thereafter, many research papers have been published on this subject and has been studied recently by many algebraists such as: Prenowitz, Corsini, Jantosciak, Leoreanu, Heideri, Davvaz, Hila, Gutan, Griffiths and Halzen.