Decomposition of intra-regular $po$-$\Gamma$-semigroups into simple components

TL;DR

This paper investigates how intra-regular $po$-$$-semigroups can be decomposed into simple components, establishing conditions for such decompositions and characterizing their structure through ideals and maximal simple subsemigroups.

Contribution

It introduces a decomposition framework for intra-regular $po$-$$-semigroups into simple components, extending previous definitions and characterizations.

Findings

Intra-regular $po$-$$-semigroups decompose into simple components.

Chains of ideals correspond to chains of simple semigroups.

Ideals being prime characterizes the structure of intra-regular $po$-$$-semigroups.

Abstract

We keep the definition of intra-regularity (left regularity) of $po$-$\Gamma$-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 $po$-$\Gamma$-semigroup into simple components. Then we prove that a $po$-$\Gamma$-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 $po$-$\Gamma$-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 $po$-$\Gamma$-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 $po$-$\Gamma$-semigroup into left simple components has been also given.