On m-quasi-ideals in m-regular ordered semigroups

TL;DR

This paper introduces and characterizes m-regular ordered semigroups and their m-quasi-ideals, establishing their properties and interrelations, and demonstrating that the set of all m-quasi-ideals forms an m-regular semigroup.

Contribution

It defines the concept of m-regularity in ordered semigroups and explores the structure of m-quasi-ideals, providing new characterizations and properties.

Findings

m-regular ordered semigroups are characterized by their m-quasi-ideals

The set of all m-quasi-ideals forms an m-regular semigroup

Interplay between different types of ideals in ordered semigroups

Abstract

In this paper we characterize left(right) ideals, bi-ideals and quasi-ideals of an ordered semigroup by an index $m$ and give some important interplays between these ideals. The concept of m-regularity of an ordered semigroups has been introduced. Moreover m-regular ordered semigroups are characterized by their m-quasi-ideals and the fact that for any m-regular ordered semigroups $A$, the set $Q_{A}$ of all m-quasi-ideals of $A$, with multiplication defined by: $Q_{1}\circ Q_{2}=(Q_{1}Q_{2}]$, for all $Q_{1},Q_{2}\in Q_{A}$, is a m-regular semigroup is obtained here.