Semigroups whose right ideals are finitely generated

TL;DR

This paper studies semigroups where all right ideals are finitely generated, providing characterizations, exploring their behavior under various constructions, and identifying conditions for specific classes to be weakly right noetherian.

Contribution

It offers new characterizations of weakly right noetherian semigroups and monoids, and analyzes their properties under quotients, subsemigroups, and direct products.

Findings

Characterization of weakly right noetherian semigroups via principal right ideals

Necessary and sufficient conditions for direct products to be weakly right noetherian

Classification of regular semigroups and semilattices with this property

Abstract

We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterise weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterise weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal{H}$ is finitely generated.