The universal semigroup of a $\Gamma$-semigroup

TL;DR

This paper introduces a universal semigroup construction for $\Gamma$-semigroups, enabling the transfer of classical semigroup results to $\Gamma$-semigroups and establishing properties like Green's theorem and simplicity across related structures.

Contribution

It constructs a universal semigroup for $\Gamma$-semigroups that preserves ideal structures and allows applying classical semigroup results to $\Gamma$-semigroups.

Findings

Green's theorem for $\Gamma$-semigroups derived from classical results

If one component is completely simple, all are

Construction simplifies analysis of $\Gamma$-semigroups

Abstract

Given a $\Gamma$-semigroup $S$, we construct a semigroup $\Sigma$ in such a way that one sided ideals and quasi-ideals of $S$ can be regarded as one sided ideals and quasi-ideals respectively of $\Sigma$. This correspondence and other properties of $\Sigma$, allow us to obtain several results for $S$ without having the need to work directly with it, but solely employing well known results of semigroup theory. For example, we obtain the Green's theorem for $\Gamma$-semigroups found in \cite{PT}, as a corollary of the usual Green's theorem in semigroups. Also we prove that, if $S$ is a $\Gamma$-semigroup and $\gamma_{0} \in \Gamma$ such that $S_{\gamma_{0}}$ is a completely simple semigroup, then for every $\gamma \in \Gamma$, $S_{\gamma}$ is completely simple too.