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.
Type
Thesis, PhD Doctor of Philosophy
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