Nil-extensions of simple and right $\pi$-inverse ordered semigroups

TL;DR

This paper investigates the structure of nil-extensions in simple and right π-inverse ordered semigroups, establishing conditions for when right π-inverse implies π-inverse and characterizing their semilattice decompositions.

Contribution

It provides a characterization of right π-inverse ordered semigroups as π-inverse in t-Archimedean contexts and describes their semilattice structure.

Findings

S is right π-inverse iff it is π-inverse in a t-Archimedean ordered semigroup

Complete semilattice decomposition of nil-extensions of these semigroups

Conditions under which the converse of right π-inverse holds

Abstract

An ordered semigroup $S$ is right $\pi$-inverse if it is $\pi$-inverse but not conversely. So the question arises under what condition the converse holds. In this paper we study nil-extensions of simple and right $\pi$-inverse ordered semigroups and prove that $S$ is right $\pi$-inverse if and only if $S$ is $\pi$-inverse in a $t$-Archimedean ordered semigroup. Moreover, we characterize complete semilattice of nil-extensions of simple and right $\pi$-inverse ordered semigroups.