On intra-regular and some left regular $\Gamma$-semigroups

TL;DR

This paper characterizes intra-regular and certain left regular $ ext{Gamma}$-semigroups using filters, showing their decomposition into simple or left simple components, and explores their structural properties.

Contribution

It provides new characterizations of intra-regular and left regular $ ext{Gamma}$-semigroups via filters and demonstrates their decomposition into simple components.

Findings

Intra-regular $ ext{Gamma}$-semigroups decompose into simple components.

Semigroups with $x ext{Gamma} M extless{}= M ext{Gamma} x$ are left regular and decompose into left simple components.

Characterizations are achieved through the use of filters.

Abstract

We characterize the intra-regular $\Gamma$-semigroups and the left regular $\Gamma$-semigroups $M$ in which $x\Gamma M\subseteq M\Gamma x$ for every $x\in M$ in terms of filters and we prove, among others, that every intra-regular $\Gamma$-semigroup is decomposable into simple components, and every $\Gamma$-semigroup $M$ for which $x\Gamma M\subseteq M\Gamma x$ is left regular, is decomposable into left simple components.