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dc.contributor.advisorRuškuc, Nik
dc.contributor.advisorLinton, Stephen
dc.contributor.authorDistler, Andreas
dc.description.abstractThe classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.en_US
dc.publisherUniversity of St Andrews
dc.rightsCreative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported
dc.subjectNilpotent semigroupsen_US
dc.subjectEnumerative combinatoricsen_US
dc.subjectData libraryen_US
dc.subjectPower group enumerationen_US
dc.subjectComputer searchen_US
dc.subjectConstraint satisfactionen_US
dc.subjectComputer algebraen_US
dc.subject.lcshCombinatorial analysisen
dc.titleClassification and enumeration of finite semigroupsen_US
dc.contributor.sponsorUniversity of St Andrewsen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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