The universal semigroup of a $\mathrm{\Gamma }$-semigroup

Given a $\mathrm{\Gamma }$-semigroup $S$, we construct a semigroup $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$. This correspondence and other properties of $\mathrm{\Sigma }$, 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 $\mathrm{\Gamma }$-semigroups found in \cite{PT}, as a corollary of the usual Green’s theorem in semigroups. Also we prove that, if $S$ is a $\mathrm{\Gamma }$-semigroup and ${\gamma }_0\in \mathrm{\Gamma }$ such that $S_{{\gamma }_0}$ is a completely simple semigroup, then for every $\gamma \in \mathrm{\Gamma }$, $S_{\gamma }$ is completely simple too.