Decomposition of intra-regular $po$-$Γ$-semigroups into simple components

Abstract

We keep the definition of intra-regularity (left regularity) of $po$-$Γ$-semigroups introduced in arXiv: 1511.00679 which is absolutely necessary for the investigation. Being able to describe the form of the elements of the principal filter by using this definition, we study the decomposition of an intra-regular $po$-$Γ$-semigroup into simple components. Then we prove that a $po$-$Γ$-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 $po$-$Γ$-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 $po$-$Γ$-semigroup $M$, the set $\{(x)_{\cal N} \mid x\in M\}$ coincides with the set of all maximal simple subsemigroups of $M$. A decomposition of left regular and left duo $po$-$Γ$-semigroup into left simple components has been also given.