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dc.contributor.advisorMitchell, James David
dc.contributor.authorWilson, Wilf A.
dc.coverage.spatial197 p.en_US
dc.description.abstractA 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.en_US
dc.publisherUniversity of St Andrews
dc.subjectSemigroup theoryen_US
dc.subjectComputational algebraen_US
dc.subjectMaximal subsemigroupsen_US
dc.subjectComputational semigroup theoryen_US
dc.subjectRees matrix semigroupsen_US
dc.subjectRees 0-matrix semigroupsen_US
dc.subjectDirect productsen_US
dc.subjectTransformation semigroupsen_US
dc.subjectDiagram monoidsen_US
dc.subjectPartition monoidsen_US
dc.subjectGenerating setsen_US
dc.subjectGreen's relationsen_US
dc.subject.lcshSemigroups--Data processingen
dc.subject.lcshAlgebra--Data processingen
dc.subject.lcshGroup theoryen
dc.titleComputational techniques in finite semigroup theoryen_US
dc.contributor.sponsorCarnegie Trust for the Universities of Scotlanden_US
dc.contributor.sponsorUniversity of St Andrews. School of Mathematics and Statisticsen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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