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dc.contributor.advisorRuškuc, Nik
dc.contributor.authorClayton, Ashley
dc.coverage.spatialvii, 171 p.en_US
dc.description.abstractSubdirect products are special types of subalgebras of direct products. The purpose of this thesis is to initiate a study of combinatorial properties of subdirect products and fiber products of semigroups and monoids, motivated by the previous work on free groups, and some recent advances in general algebra. In Chapter 1, we outline the necessary preliminary definitions and results, including elements of algebraic semigroup theory, formal language theory, automata theory and universal algebra. In Chapter 2, we consider the number of subsemigroups and subdirect products of ℕ𝗑ℕ up to isomorphism. We obtain uncountably many such objects, and characterise the finite semigroups 𝘚 for which ℕ𝗑𝘚 has uncountable many subsemigroups and subdirect products up to isomorphism. In Chapter 3, we consider particular finite generating sets for subdirect products of free semigroups introduced as "sets of letter pairs". We classify and count these sets which generate subdirect and fiber products, and discuss their abundance. In Chapter 4, we consider finite generation and presentation for fiber products of free semigroups and monoids over finite fibers. We give a characterisation for finite generation of the fiber product of two free monoids over a finite fiber, and show that this implies finite presentation. We show that the fiber product of two free semigroups over a finite fiber is never finitely generated, and obtain necessary conditions on an infinite fiber for finite generation. In Chapter 5, we consider the problem of finite generation for fiber products of free semigroups and monoids over a free fiber. We construct two-tape automata which we use to determine the language of indecomposable elements of the fiber product, which algorithmically decides when they are finitely generated. Finally in Chapter 6, we summarise our findings, providing some further questions based on the results of the thesis.en_US
dc.description.sponsorship"I would like to thank the EPSRC (grant number EP/N509759/1) for their generous funding and skills training. I would also like to thank the entire School of Mathematics & Statistics of the University of St Andrews for their nancial support..." -- Acknowledgementsen
dc.publisherUniversity of St Andrews
dc.subjectSubdirect productsen_US
dc.subjectFiber productsen_US
dc.subject.lcshGroup theoryen
dc.titleSubdirect products of free semigroups and monoidsen_US
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en_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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