Left equalizer simple semigroups

TL;DR

This paper characterizes and constructs a class of semigroups called left equalizer simple semigroups, focusing on their algebraic properties and the structure of related semigroup algebras and congruences.

Contribution

It provides a characterization and construction method for left equalizer simple semigroups, expanding understanding of their algebraic structure and associated ideals.

Findings

Characterization of left equalizer simple semigroups

Construction methods for these semigroups

Analysis of related semigroup algebra ideals

Abstract

In this paper we characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence $\varrho$ on a semigroup $S$, let ${\mathbb F}[\varrho]$ denote the ideal of the semigroup algebra ${\mathbb F}[S]$ which determines the kernel of the extended homomorphism of ${\mathbb F}[S]$ onto ${\mathbb F}[S/\varrho]$ induced by the canonical homomorphism of $S$ onto $S/\varrho$. We examine the right colons $({\mathbb F}[\varrho]:_r{\mathbb F}[S])=\{ a\in {\mathbb F}[S]:\ {\mathbb F}[S]a\subseteq {\mathbb F}[\varrho ]\}$ in general, and in that special case when $\varrho$ has the property that the factor semigroup $S/\varrho$ is left equalizer simple.