The ascending chain condition on principal right ideals for semigroup constructions

TL;DR

This paper studies the property of being noetherian in semigroups, which involves the ascending chain condition on principal right ideals, and examines how this property behaves under various semigroup constructions.

Contribution

It characterizes the preservation of noetherian property in semigroups through standard constructions like semidirect products and Rees matrix semigroups.

Findings

noetherian property is preserved under certain semigroup constructions.

Conditions for noetherian property to hold in complex semigroup structures.

Insights into the structure of noetherian semigroups and their extensions.

Abstract

We call a semigroup $\mathcal{R}$-noetherian if it satisfies the ascending chain condition on principal right ideals, or, equivalently, the ascending chain condition on $\mathcal{R}$-classes. We investigate the behaviour of the property of being $\mathcal{R}$-noetherian under the following standard semigroup-theoretic constructions: semidirect products, Sch\"utzenberger products, free products, Rees matrix semigroups, Brandt extensions, Bruck-Reilly extensions and semilattices of semigroups.