Reflexive Unitary Subsemigroups of Left Simple Semigroups

TL;DR

This paper introduces reflexive unitary subsemigroups of left simple semigroups to generalize group-theoretic concepts like normal and composition series, addressing limitations of ideal series in such semigroups.

Contribution

It identifies reflexive unitary subsemigroups as a new tool to extend group theory notions to left simple semigroups, filling a gap in semigroup theory.

Findings

Reflexive unitary subsemigroups facilitate the generalization of group series concepts.

New structural properties of left simple semigroups are established.

The approach offers a framework for applying group theory methods to semigroups.

Abstract

Ideal series of semigroups play an important role in the examination of semigroups which have proper two-sided ideals. But the corresponding theorems cannot be used when left simple (or right simple or simple) semigroups are considered. So it is a natural idea that we try to use the group theoretical methods (instead of the ring theoretical ones) for the examination of these semigroups. The purpose of this paper is to find a suitable type of subsemigroups of left simple semigroups which makes possible for us to generalize some notions (the notion of the normal series and the composition series of groups) and some results concerning the groups to the left simple semigroups. We note that the subsemigroups we are looking for are the reflexive unitary subsemigroups of left simple semigroups.