Finiteness conditions on semigroups relating to their actions and one-sided congruences

Abstract

The purpose of this thesis is threefold: firstly, to develop a systematic theory of presentations of monoid acts; secondly, to study finiteness conditions on semigroups relating to finite generation of one-sided congruences; and thirdly, to establish connections between each of these finiteness conditions, restricted to the class of monoids, with finite presentability of acts.

We find general presentations for various monoid act constructions/components, leading to a number of finite presentability results. In particular, we consider subacts, Rees quotients, unions of subacts, direct products and wreath products. A semigroup 𝑆 is called right noetherian if every right congruence on 𝑆 is finitely generated. We present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.

Finally, we introduce and study the condition that every right congruence of finite index on a semigroup is finitely generated. We call semigroups satisfying this condition f-noetherian. It turns out that every finitely generated semigroup is f-noetherian. We investigate, for various semigroup classes, whether the property of being f-noetherian coincides with finite generation.