The universal semigroup of a $Γ$-semigroup

Abstract

Given a $Γ$-semigroup $S$, we construct a semigroup $Σ$ 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 $Σ$. This correspondence and other properties of $Σ$, 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 $Γ$-semigroups found in \cite{PT}, as a corollary of the usual Green's theorem in semigroups. Also we prove that, if $S$ is a $Γ$-semigroup and $γ_{0} \in Γ$ such that $S_{γ_{0}}$ is a completely simple semigroup, then for every $γ\in Γ$, $S_γ$ is completely simple too.