Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.759374
Title: Computational techniques in finite semigroup theory
Author: Wilson, Wilf A.
ISNI:       0000 0004 7431 4141
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 2019
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Abstract:
A semigroup is simply a set with an associative binary operation; computational semigroup theory is the branch of mathematics concerned with developing techniques for computing with semigroups, as well as investigating semigroups with the help of computers. This thesis explores both sides of computational semigroup theory, across several topics, especially in the finite case. The central focus of this thesis is computing and describing maximal subsemigroups of finite semigroups. A maximal subsemigroup of a semigroup is a proper subsemigroup that is contained in no other proper subsemigroup. We present novel and useful algorithms for computing the maximal subsemigroups of an arbitrary finite semigroup, building on the paper of Graham, Graham, and Rhodes from 1968. In certain cases, the algorithms reduce to computing maximal subgroups of finite groups, and analysing graphs that capture information about the regular I-classes of a semigroup. We use the framework underpinning these algorithms to describe the maximal subsemigroups of many families of finite transformation and diagram monoids. This reproduces and greatly extends a large amount of existing work in the literature, and allows us to easily see the common features between these maximal subsemigroups. This thesis is also concerned with direct products of semigroups, and with a special class of semigroups known as Rees 0-matrix semigroups. We extend known results concerning the generating sets of direct products of semigroups; in doing so, we propose techniques for computing relatively small generating sets for certain kinds of direct products. Additionally, we characterise several features of Rees 0-matrix semigroups in terms of their underlying semigroups and matrices, such as their Green's relations and generating sets, and whether they are inverse. In doing so, we suggest new methods for computing Rees 0-matrix semigroups.
Supervisor: Mitchell, James David Sponsor: Carnegie Trust for the Universities of Scotland ; University of St Andrews
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.759374  DOI: Not available
Keywords: Semigroup theory ; Computational algebra ; Maximal subsemigroups ; Semigroups ; Computational semigroup theory ; Rees matrix semigroups ; Rees 0-matrix semigroups ; Direct products ; Algorithms ; Transformation semigroups ; Diagram monoids ; Partition monoids ; Monoids ; Generating sets ; Green's relations ; QA182.W5 ; Semigroups--Data processing ; Algebra--Data processing ; Group theory
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