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dc.contributor.advisorRuškuc, Nik
dc.contributor.advisorBleak, Collin Patrick
dc.contributor.authorMcLeman, Alexander Lewis Andrew
dc.coverage.spatial167en_US
dc.date.accessioned2015-04-23T14:46:35Z
dc.date.available2015-04-23T14:46:35Z
dc.date.issued2015-06-26
dc.identifieruk.bl.ethos.644842
dc.identifier.urihttps://hdl.handle.net/10023/6558
dc.description.abstractLet S be a semigroup, C(S) the automaton constructed from the right Cayley graph of S with respect to all of S as the generating set and ∑(C(S)) the automaton semigroup constructed from C(S). Such semigroups are termed Cayley automaton semigroups. For a given semigroup S we aim to establish connections between S and ∑(C(S)). For a finite monogenic semigroup S with a non-trivial cyclic subgroup C[sub]n we show that ∑(C(S)) is a small extension of a free semigroup of rank n, and that in the case of a trivial subgroup ∑(C(S)) is finite. The notion of invariance is considered and we examine those semigroups S satisfying S ≅ ∑(C(S)). We classify which bands satisfy this, showing that they are those bands with faithful left-regular representations, but exhibit examples outwith this classification. In doing so we answer an open problem of Cain. Following this, we consider iterations of the construction and show that for any n there exists a semigroup where we can iterate the construction n times before reaching a semigroup satisfying S ≅ ∑(C(S)). We also give an example of a semigroup where repeated iteration never produces a semigroup satisfying S ≅ ∑(C(S)). Cayley automaton semigroups of infinite semigroups are also considered and we generalise and extend a result of Silva and Steinberg to cancellative semigroups. We also construct the Cayley automaton semigroup of the bicyclic monoid, showing in particular that it is not finitely generated.en_US
dc.language.isoenen_US
dc.publisherUniversity of St Andrews
dc.subjectSemigroupen_US
dc.subjectAutomatonen_US
dc.subject.lccQA182.M6
dc.subject.lcshSemigroupsen_US
dc.subject.lcshCayley graphsen_US
dc.titleCayley automaton semigroupsen_US
dc.typeThesisen_US
dc.type.qualificationlevelDoctoralen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US


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