Interior ideal in regular and intra regular semigroup

TL;DR

This paper investigates interior ideals in regular and intra-regular semigroups, introduces new classes of interior ideals, and characterizes semigroups based on their interior ideal properties.

Contribution

It introduces and characterizes various classes of interior ideals in semigroups, including strongly prime and irreducible interior ideals, and explores their interplay with other ideals.

Findings

Characterization of semigroups with strongly prime interior ideals

Introduction of strongly prime, semiprime, and irreducible interior ideals

Descriptions of semigroups via minimal interior ideals

Abstract

Following G.Szasz [2] a subsemigroup I of semigroup S is called an interior ideal if SIS \subset I. In this paper we explore the classes of regular semigroup and its different subclasses by their interior ideals. Furthermore, we introduce the strongly prime, prime, semiprime, strongly irreducible, and irreducible interior ideals of semigroups and also characterize those semigroups for which each interior ideal is strongly prime. Some important interplay between the classes of all interior ideals and other ideals are given here. In addition to this, we present different characterizations of semigroups by their minimal interior ideals.