On $\pi$-extensions of the semigroup $\mathbb{Z}_+$

TL;DR

This paper investigates inverse and non-inverse $ ext{pi}$-extensions of the semigroup $ ext{Z}_+$, establishing conditions for when these extensions are inverse and demonstrating the existence of non-inverse extensions.

Contribution

It characterizes when $ ext{pi}$-extensions of $ ext{Z}_+$ are inverse and proves the existence of non-inverse $ ext{pi}$-extensions, advancing understanding of their structure.

Findings

$ ext{pi}$-extension of $ ext{Z}_+$ is inverse iff it coincides with $ ext{pi}( ext{Z}_+)$

Existence of non-inverse $ ext{pi}$-extensions for $ ext{Z}_+$

Conditions for inverse $ ext{pi}$-extensions established

Abstract

We study inverse $\pi$-extensions of the semigroup $\mathbb{Z}_+$. It is shown that $\pi$-extension of the semigroup $\mathbb{Z}_+$ is inverse, iff its $\pi$-extension coincides with $\pi(\mathbb{Z}_+)$. The existence of a non-inverse $\pi$-extension for semigroup $\mathbb{Z}_+$ is proved.