Generation and presentations of semigroup constructions : Bruck-Reilly extensions and P-semigroups

Abstract

In this thesis we study problems regarding finite presentability of Bruck-Reilly extensions, finite generation of the underlying monoids, and finite generation of P-unitary inverse semigroups. The first main question we consider is: Let M be a monoid and θ and endomorphism of M. If the Bruck-Reilly extension BR(M, θ) is finitely presented is the monoid M necessarily finitely generated? We answer this question for the following classes of monoids: semilattices; Clifford monoids; zero monoids; free monoids; completely (0-)simple semigroups; and semidirect products of semilattices by groups. This allows us to obtain necessary and sufficient conditions for the Bruck-Reilly extensions of these classes of monoids to be finitely presented. We also show that, like the free inverse monoid, a Bruck-Reilly extension (of an inverse monoid) is not necessarily finitely presented as a monoid when it happens to be finitely presented as an inverse monoid. We then consider the question: When are P-semigroups, or P-unitary inverse semigroups, finitely generated? We give necessary and sufficient conditions for a P-semigroup P(G, X, Y) to be finitely generated in the case when X\Y is finite, and consider several particular cases when X\Y is infinite.