Local embeddability and related properties of semigroups

Abstract

After being firmly established as a field of mathematical study separate from group theory, semigroup theory has been in active development throughout recent years. Many of its important results stem from generalising corresponding work in groups, but at the same time a significant progress has been made in introducing and applying concepts specifically designed for semigroups or subclasses of semigroups distinct from groups. The present work deals with two topics of semigroup theory, namely local embeddability into finite semigroups (LEF) and lengths of semigroup algebras. Both of them share a connection to a wider area of finite and finitary algebraic structures as well as to combinatorics, and have been previously examined in detail for corresponding group cases. Following the general pattern, in this thesis we present results which generalise ideas established for the group case and develop new semigroup methods and notions intended for a deeper understanding of the topic. We begin with an overview of several relevant subjects, including formal language theory and model theory, and then delve into properties of LEF semigroups, inspecting equivalent definitions, connections to groups, structural invariants and important examples of LEF and non-LEF behaviour. Using this framework, we also consider the class of inverse semigroups, look into the taxonomy of formal languages related to LEF semigroups via syntactic congruence and introduce and examine a new class of semigroups locally wrapped by finite structures (LWF). We finish the text with results on algebra lengths, proving several bounds for general and inverse cases and computing lengths of Tsetlin library algebras for low dimensions.

Date of Award	23 Jun 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Marianne Johnson (Supervisor) & Mark Kambites (Supervisor)