The adjoint semigroup of a $\Gamma$-semigroup

Elton Pasku

Given a $\Gamma$-semigroup $S$ and a fixed $\gamma_{0} \in \Gamma$, we construct a semigroup $\Sigma_{\gamma_{0}}$ in such a way that there is a one to one correspondence between the set of principal one sided ideals (resp. principal quasi-ideals) of $S$ and their counterparts in $\Sigma_{\gamma_{0}}$. This correspondence allows us to obtain several results for $S$ without having the need to work directly with it, but working with $\Sigma_{\gamma_{0}}$ instead and employing well known results of semigroup theory. For example, we obtain an analogue of the Green's theorem for $\Gamma$-semigroups 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.