On Intra-regular Ordered Gamma-semigroups

Abstract

We study the decomposition of an intra-regular $po$-$\mathrm{\Gamma }$-semigroup into simple components. Then we prove that a $po$-$\mathrm{\Gamma }$-semigroup $M$ is intra-regular and the ideals of $M$ form a chain if and only if $M$ is a chain of simple semigroups. Moreover, a $po$-$\mathrm{\Gamma }$-semigroup $M$ is intra-regular and the ideals of $M$ form a chain if and only if the ideals of $M$ are prime. Finally, for an intra-regular $po$-$\mathrm{\Gamma }$-semigroup $M$, the set $\{(x{)}_{\mathcal{N}}\mid x\in M\}$ coincides with the set of all maximal simple subsemigroups of $M$. A decomposition of left regular and left duo $po$-$\mathrm{\Gamma }$-semigroup into left simple components has been also given.