Pure Mathematics Theses http://hdl.handle.net/10023/99 2022-10-01T14:49:52Z Base size and generating graphs of primitive permutation groups http://hdl.handle.net/10023/25826 In this thesis we consider base size and properties of the generating graph for finite groups. Let Ω = {1,...,n}, let Sₙ = Sym({1,...,n}) and let G ≤ Sₙ. A base for G is a sequence Λ = (ω₁, . . . , ωₖ) of points in Ω such that the pointwise stabilizer, G_{ω₁,...,ωₖ} , is the identity. The base size of G, denoted by b(G, Ω) or b(G), is the length of the shortest base. We say that Λ is an irredundant base if G > G_{ω₁} > G_{ω₁,ω₂} > ··· > G_{ω₁,ω₂,...,ωₖ} = 1. If no irredundant base is longer than Λ, then we say that Λ is a maximal irredundant base for G and denote its length by I(G). A group is called large base if it is either a product action or almost simple group, and its socle is one or more copies of the alternating group Aᵣ acting on k-sets. Let G be a primitive subgroup of Sₙ that is not large base. We prove that any irredundant base for G has size at most 5log₂n. This bound is best possible up to a small multiplicative constant and is the first logarithmic bound on the size of an irredundant base for such groups. We show that for any constant c, there are infinitely many primitive groups with maximal irredundant base size at least c times the minimal base size. As a corollary of the first result, the relational complexity of G, denoted RC(G) (see Definition 2.2.10), is at most 5log₂n + 1. In addition the maximal size of a minimal base and the height, denoted B(G) and H(G) (see Definitions 2.2.1 and 2.2.5), are both at most 5log₂n. Furthermore, we deduce that a base for G of size at most 5log₂n can be computed in polynomial time. The generating graph Γ(G) of a finite group G has vertex set the non-identity elements of G, with two elements connected exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ(G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ(G), but this coclique need not be maximal. Let G = Sₙ or Aₙ. We first determine when the intransitive maximal subgroups of G are maximal cocliques in Γ(G), and when they are not we find the unique maximal coclique in which they are contained. We then show that for sufficiently large n, the imprimitive maximal subgroups of G are all maximal cocliques in Γ(G). In addition, using the result on intransitive maximal subgroups we prove that a conjecture of Cameron, Lucchini, and Roney-Dougal holds for G under certain restrictions on n. Namely we prove that two elements of G have identical sets of neighbours in Γ(G) if and only if they belong to exactly the same maximal subgroups. Finally under another set of restrictions on n we then determine precisely which maximal subgroups are maximal cocliques in Γ(G). 2022-06-14T00:00:00Z Kelsey, Veronica In this thesis we consider base size and properties of the generating graph for finite groups. Let Ω = {1,...,n}, let Sₙ = Sym({1,...,n}) and let G ≤ Sₙ. A base for G is a sequence Λ = (ω₁, . . . , ωₖ) of points in Ω such that the pointwise stabilizer, G_{ω₁,...,ωₖ} , is the identity. The base size of G, denoted by b(G, Ω) or b(G), is the length of the shortest base. We say that Λ is an irredundant base if G > G_{ω₁} > G_{ω₁,ω₂} > ··· > G_{ω₁,ω₂,...,ωₖ} = 1. If no irredundant base is longer than Λ, then we say that Λ is a maximal irredundant base for G and denote its length by I(G). A group is called large base if it is either a product action or almost simple group, and its socle is one or more copies of the alternating group Aᵣ acting on k-sets. Let G be a primitive subgroup of Sₙ that is not large base. We prove that any irredundant base for G has size at most 5log₂n. This bound is best possible up to a small multiplicative constant and is the first logarithmic bound on the size of an irredundant base for such groups. We show that for any constant c, there are infinitely many primitive groups with maximal irredundant base size at least c times the minimal base size. As a corollary of the first result, the relational complexity of G, denoted RC(G) (see Definition 2.2.10), is at most 5log₂n + 1. In addition the maximal size of a minimal base and the height, denoted B(G) and H(G) (see Definitions 2.2.1 and 2.2.5), are both at most 5log₂n. Furthermore, we deduce that a base for G of size at most 5log₂n can be computed in polynomial time. The generating graph Γ(G) of a finite group G has vertex set the non-identity elements of G, with two elements connected exactly when they generate G. A coclique in a graph is an empty induced subgraph, so a coclique in Γ(G) is a subset of G such that no pair of elements generate G. A coclique is maximal if it is contained in no larger coclique. It is easy to see that the non-identity elements of a maximal subgroup of G form a coclique in Γ(G), but this coclique need not be maximal. Let G = Sₙ or Aₙ. We first determine when the intransitive maximal subgroups of G are maximal cocliques in Γ(G), and when they are not we find the unique maximal coclique in which they are contained. We then show that for sufficiently large n, the imprimitive maximal subgroups of G are all maximal cocliques in Γ(G). In addition, using the result on intransitive maximal subgroups we prove that a conjecture of Cameron, Lucchini, and Roney-Dougal holds for G under certain restrictions on n. Namely we prove that two elements of G have identical sets of neighbours in Γ(G) if and only if they belong to exactly the same maximal subgroups. Finally under another set of restrictions on n we then determine precisely which maximal subgroups are maximal cocliques in Γ(G). On singular pencils of matrices http://hdl.handle.net/10023/24945 "This thesis is a study of Singular Matrix Pencils under various aspects. In part (I) a new derivation of the Canonical Form of matrix pencils is given. This suggests investigation of the transformations of a pencil into itself (part (II)). Finally, part (III) deals with the canonical form of singular pencils of special types, namely those whose members are induced (or invariant) matrices." -- Preface 1936-01-01T00:00:00Z Ledermann, Walter "This thesis is a study of Singular Matrix Pencils under various aspects. In part (I) a new derivation of the Canonical Form of matrix pencils is given. This suggests investigation of the transformations of a pencil into itself (part (II)). Finally, part (III) deals with the canonical form of singular pencils of special types, namely those whose members are induced (or invariant) matrices." -- Preface The invariant theory of linear complexes associated with a quaternary quadric http://hdl.handle.net/10023/24930 "The invariants and covariants of a quaternary system of a linear complex and a quadric have been discussed by Weitzenbock, but he excludes the mixed concomitants. In the present paper the concomitants (which will include mixed forms) of a quadric and two linear complexes are discussed. The Reduced Prepared System is given in §3, the Complete System in §8 and some invariants of the covariant forms in §9. The concomitants of a quadric and 𝑛 linear complexes have next been considered and the corresponding reduced system of typical forms, is given at the end of the paper." -- From the Introduction. 1929-01-01T00:00:00Z DasGupta, Pramathanath "The invariants and covariants of a quaternary system of a linear complex and a quadric have been discussed by Weitzenbock, but he excludes the mixed concomitants. In the present paper the concomitants (which will include mixed forms) of a quadric and two linear complexes are discussed. The Reduced Prepared System is given in §3, the Complete System in §8 and some invariants of the covariant forms in §9. The concomitants of a quadric and 𝑛 linear complexes have next been considered and the corresponding reduced system of typical forms, is given at the end of the paper." -- From the Introduction. Studies in form-theory : 1. Mixed determinants - 2. The pedal correspondence http://hdl.handle.net/10023/24896 1924-01-01T00:00:00Z Vaidyanathaswamy, R. Contributions to the theory of apolarity http://hdl.handle.net/10023/23956 1924-01-01T00:00:00Z Vaidyanathaswamy, R. Multifractal measures : from self-affine to nonlinear http://hdl.handle.net/10023/23786 This thesis is based on three papers the author wrote during his time as a PhD student [28, 17, 33]. In Chapter 2 we study 𝐿[sup]𝑞-spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we find counterexamples to a statement of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the 𝐿[sup]𝑞-spectrum. As a further application we provide examples of self-affine measures whose 𝐿[sup]𝑞-spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the 𝐿[sup]𝑞-spectra, which in certain cases yield sharp results. In Chapter 3 we study 𝐿[sup]𝑞-spectra of measures in the plane generated by certain nonlinear maps. In particular we study attractors of iterated function systems consisting of maps whose components are 𝐶[sup](1+α) and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the 𝐿[sup]𝑞-spectrum of Bernoulli measures supported on such sets using an appropriately defined analogue of the singular value function and an appropriate pressure function. In Chapter 4 we study a more general class of invariant measures supported on the attractors introduced in Chapter 3. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula. 2021-12-01T00:00:00Z Lee, Lawrence David This thesis is based on three papers the author wrote during his time as a PhD student [28, 17, 33]. In Chapter 2 we study 𝐿[sup]𝑞-spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we find counterexamples to a statement of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures. We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the 𝐿[sup]𝑞-spectrum. As a further application we provide examples of self-affine measures whose 𝐿[sup]𝑞-spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the 𝐿[sup]𝑞-spectra, which in certain cases yield sharp results. In Chapter 3 we study 𝐿[sup]𝑞-spectra of measures in the plane generated by certain nonlinear maps. In particular we study attractors of iterated function systems consisting of maps whose components are 𝐶[sup](1+α) and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the 𝐿[sup]𝑞-spectrum of Bernoulli measures supported on such sets using an appropriately defined analogue of the singular value function and an appropriate pressure function. In Chapter 4 we study a more general class of invariant measures supported on the attractors introduced in Chapter 3. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula. Enumerating 0-simple semigroups http://hdl.handle.net/10023/23558 Computational semigroup theory involves the study and implementation of algorithms to compute with semigroups. Efficiency is of central concern and often follows from the insight of semigroup theoretic results. In turn, computational methods allow for analysis of semigroups which can provide intuition leading to theoretical breakthroughs. More efficient algorithms allow for more cases to be computed and increases the potential for insight. In this way, research into computational semigroup theory and abstract semigroup theory forms a feedback loop with each benefiting the other. In this thesis the primary focus will be on counting isomorphism classes of finite 0-simple semigroups. These semigroups are in some sense the building blocks of finite semigroups due to their correspondence with the Greens 𝒟-classes of a semigroup. The theory of Rees 0-matrix semigroups links these semigroups to matrices with entries from 0-groups. Special consideration will be given to the enumeration of certain sub-cases, most prominently the case of congruence free semigroups. The author has implemented these enumeration techniques and applied them to count isomorphism classes of 0-simple semigroups and congruence free semigroups by order. Included in this thesis are tables of the number of 0-simple semigroups of orders less than or equal to 130, up to isomorphism. Also included are tables of the numbers of congruence free semigroups, up to isomorphism, with m Green’s ℒ-classes and n Green’s ℛ-classes for all mn less than or equal to 100, as well as for various other values of m,n. Furthermore a database of finite 0-simple semigroups has been created and we detail how this was done. The implementation of these enumeration methods and the database are publicly available as GAP code. In order to achieve these results pertaining to finite 0-simple semigroups we invoke the theory of group actions and prove novel combinatorial results. Most notably, we have deduced formulae for enumerating the number of binary matrices with distinct rows and columns up to row and column permutations. There are also two sections dedicated to covers of E-unitary inverse semigroups, and presentations of factorisable orthodox monoids, respectively. In the first, we explore the concept of a minimal E-unitary inverse cover, up to isomorphism, by defining various sensible orderings. We provide examples of Clifford semigroups showing that, in general, these orderings do not have a unique minimal element. Finally, we pose conjectures about the existence of unique minimal E-unitary inverse covers of Clifford semigroups, when considered up to an equivalence weaker than isomorphism. In the latter section, we generalise the theory of presentations of factorisable inverse monoids to the more general setting of factorisable orthodox monoids. These topics were explored early in the authors doctoral studies but ultimately in less depth than the research on 0-simple semigroups. 2021-06-29T00:00:00Z Russell, Christopher Computational semigroup theory involves the study and implementation of algorithms to compute with semigroups. Efficiency is of central concern and often follows from the insight of semigroup theoretic results. In turn, computational methods allow for analysis of semigroups which can provide intuition leading to theoretical breakthroughs. More efficient algorithms allow for more cases to be computed and increases the potential for insight. In this way, research into computational semigroup theory and abstract semigroup theory forms a feedback loop with each benefiting the other. In this thesis the primary focus will be on counting isomorphism classes of finite 0-simple semigroups. These semigroups are in some sense the building blocks of finite semigroups due to their correspondence with the Greens 𝒟-classes of a semigroup. The theory of Rees 0-matrix semigroups links these semigroups to matrices with entries from 0-groups. Special consideration will be given to the enumeration of certain sub-cases, most prominently the case of congruence free semigroups. The author has implemented these enumeration techniques and applied them to count isomorphism classes of 0-simple semigroups and congruence free semigroups by order. Included in this thesis are tables of the number of 0-simple semigroups of orders less than or equal to 130, up to isomorphism. Also included are tables of the numbers of congruence free semigroups, up to isomorphism, with m Green’s ℒ-classes and n Green’s ℛ-classes for all mn less than or equal to 100, as well as for various other values of m,n. Furthermore a database of finite 0-simple semigroups has been created and we detail how this was done. The implementation of these enumeration methods and the database are publicly available as GAP code. In order to achieve these results pertaining to finite 0-simple semigroups we invoke the theory of group actions and prove novel combinatorial results. Most notably, we have deduced formulae for enumerating the number of binary matrices with distinct rows and columns up to row and column permutations. There are also two sections dedicated to covers of E-unitary inverse semigroups, and presentations of factorisable orthodox monoids, respectively. In the first, we explore the concept of a minimal E-unitary inverse cover, up to isomorphism, by defining various sensible orderings. We provide examples of Clifford semigroups showing that, in general, these orderings do not have a unique minimal element. Finally, we pose conjectures about the existence of unique minimal E-unitary inverse covers of Clifford semigroups, when considered up to an equivalence weaker than isomorphism. In the latter section, we generalise the theory of presentations of factorisable inverse monoids to the more general setting of factorisable orthodox monoids. These topics were explored early in the authors doctoral studies but ultimately in less depth than the research on 0-simple semigroups. Coincidence and disparity of fractal dimensions http://hdl.handle.net/10023/23381 We investigate the dimension and structure of four fractal families: inhomogeneous attractors, fractal projections, fractional Brownian images, and elliptical polynomial spirals. For each family, particular attention is given to the relationships between different notions of dimension. This may take the form of determining conditions for them to coincide, or, in the case they differ, calculating the spectrum of dimensions interpolating between them. Material for this thesis is drawn from the papers [6,7,8,9,10]. First, we develop the dimension theory of inhomogeneous attractors for non-linear and affine iterated function systems. In both cases, we find natural quantities that bound the upper box-counting dimension from above and identify sufficient conditions for these bounds to be obtained. Our work improves and unifies previous theorems on inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets. Second, we prove that the intermediate dimensions of the orthogonal projection of a Borel set 𝐸 ⸦ ℝⁿ onto a linear subspace 𝑉 are almost surely independent of the choice of subspace. Similar methods identify the almost sure value of the dimension of Borel sets under index-α fractional Brownian motion. Various applications are given, including a surprising result that relates the box dimension of the Hölder images of a set to the Hausdorff dimension of the preimages. Finally, we investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis, computing explicitly their intermediate, box-counting and Assouad-type dimensions. Relying on this, we bound the Hölder regularity of maps that deform one spiral into another, generalising the winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents. 2021-06-29T00:00:00Z Burrell, Stuart Andrew We investigate the dimension and structure of four fractal families: inhomogeneous attractors, fractal projections, fractional Brownian images, and elliptical polynomial spirals. For each family, particular attention is given to the relationships between different notions of dimension. This may take the form of determining conditions for them to coincide, or, in the case they differ, calculating the spectrum of dimensions interpolating between them. Material for this thesis is drawn from the papers [6,7,8,9,10]. First, we develop the dimension theory of inhomogeneous attractors for non-linear and affine iterated function systems. In both cases, we find natural quantities that bound the upper box-counting dimension from above and identify sufficient conditions for these bounds to be obtained. Our work improves and unifies previous theorems on inhomogeneous self-affine carpets, while providing inhomogeneous analogues of Falconer's seminal results on homogeneous self-affine sets. Second, we prove that the intermediate dimensions of the orthogonal projection of a Borel set 𝐸 ⸦ ℝⁿ onto a linear subspace 𝑉 are almost surely independent of the choice of subspace. Similar methods identify the almost sure value of the dimension of Borel sets under index-α fractional Brownian motion. Various applications are given, including a surprising result that relates the box dimension of the Hölder images of a set to the Hausdorff dimension of the preimages. Finally, we investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis, computing explicitly their intermediate, box-counting and Assouad-type dimensions. Relying on this, we bound the Hölder regularity of maps that deform one spiral into another, generalising the winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents. Subdirect products of free semigroups and monoids http://hdl.handle.net/10023/21333 Subdirect 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. 2020-12-01T00:00:00Z Clayton, Ashley Subdirect 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. On the regularity dimensions of measures http://hdl.handle.net/10023/20218 This body of work is based upon the following three papers that the author wrote during his PhD with Jonathan Fraser and Han Yu: [FH20, HY17, How19]. Chapter 1 starts by introducing many of the common tools and notation that will be used throughout this thesis. This will cover the main notions of dimensions discussed from both the set and the measure perspectives. An emphasis will be placed on their relationships where possible. This will provide a solid base upon which to expand. Many of the standard results in this part can be found in fractal geometry textbooks such as [Fal03, Mat95] if further reading was desired. The first results discussed in Chapter 2 will cover some of the regularity dimensions’ properties such as general bounds in relation to the Assouad and lower dimensions, local dimensions and the Lq-spectrum. The Assoaud and lower dimensions are known to interact pleasantly with weak tangents and these ideas are discussed in the regularity dimension setting. We then calculate the regularity dimensions for several specific example measures such as self-similar and self-affine measures which provides an opportunity to discuss the sharpness of the previously obtained bounds. This work originates in [FH20] where the upper regularity dimension was studied, with many of the lower regularity dimension results being natural extensions. In Chapter 3 we continue the study of the upper and lower regularity dimensions with an emphasis on how they can be used to quantify doubling and uniform perfectness of measures. This starts with an explicit relation between the upper regularity dimension and the doubling constants along with a similar link between the lower regularity dimension and the constants of uniform perfectness. We then turn our attention to a technical result which can be made more quantitative thanks to the regularity dimensions. It is interesting to study how properties, such as doubling, change under pushforwards by different types of maps, here we study the regularity dimensions of pushforward measures with respect to quasisymmetric homeomorphisms. We round this chapter out with an interesting application of the lower regularity to Diophantine approximation by noting the equivalence between uniform perfectness and weakly absolutely α-decaying measures. The original material for this part can be found in [How19] with part of the first section integrating a result of [FH20]. Finally, in Chapter 4, we will consider graphs of Brownian motion, and more generally, graphs of Levy processes. This will involve the calculation of the lower and Assouad dimensions for such sets and then the regularity dimensions of measures pushed onto these graphs from the real line. These graphs are the only examples in this thesis for which the Assouad and lower dimensions had not been previously calculated so we delve deeper into the area, studying graphs of functions defined as stochastic integrals as well. This chapter is based on the paper [HY17] for the set theoretic half, with the regularity dimension results coming from [How19]. 2020-01-01T00:00:00Z Howroyd, Douglas Charles This body of work is based upon the following three papers that the author wrote during his PhD with Jonathan Fraser and Han Yu: [FH20, HY17, How19]. Chapter 1 starts by introducing many of the common tools and notation that will be used throughout this thesis. This will cover the main notions of dimensions discussed from both the set and the measure perspectives. An emphasis will be placed on their relationships where possible. This will provide a solid base upon which to expand. Many of the standard results in this part can be found in fractal geometry textbooks such as [Fal03, Mat95] if further reading was desired. The first results discussed in Chapter 2 will cover some of the regularity dimensions’ properties such as general bounds in relation to the Assouad and lower dimensions, local dimensions and the Lq-spectrum. The Assoaud and lower dimensions are known to interact pleasantly with weak tangents and these ideas are discussed in the regularity dimension setting. We then calculate the regularity dimensions for several specific example measures such as self-similar and self-affine measures which provides an opportunity to discuss the sharpness of the previously obtained bounds. This work originates in [FH20] where the upper regularity dimension was studied, with many of the lower regularity dimension results being natural extensions. In Chapter 3 we continue the study of the upper and lower regularity dimensions with an emphasis on how they can be used to quantify doubling and uniform perfectness of measures. This starts with an explicit relation between the upper regularity dimension and the doubling constants along with a similar link between the lower regularity dimension and the constants of uniform perfectness. We then turn our attention to a technical result which can be made more quantitative thanks to the regularity dimensions. It is interesting to study how properties, such as doubling, change under pushforwards by different types of maps, here we study the regularity dimensions of pushforward measures with respect to quasisymmetric homeomorphisms. We round this chapter out with an interesting application of the lower regularity to Diophantine approximation by noting the equivalence between uniform perfectness and weakly absolutely α-decaying measures. The original material for this part can be found in [How19] with part of the first section integrating a result of [FH20]. Finally, in Chapter 4, we will consider graphs of Brownian motion, and more generally, graphs of Levy processes. This will involve the calculation of the lower and Assouad dimensions for such sets and then the regularity dimensions of measures pushed onto these graphs from the real line. These graphs are the only examples in this thesis for which the Assouad and lower dimensions had not been previously calculated so we delve deeper into the area, studying graphs of functions defined as stochastic integrals as well. This chapter is based on the paper [HY17] for the set theoretic half, with the regularity dimension results coming from [How19]. Orderings on words and permutations http://hdl.handle.net/10023/18465 Substructure orderings are ubiquitous throughout combinatorics and all of mathematics. In this thesis we consider various orderings on words, as well as the consecutive involvement ordering on permutations. Throughout there will be a focus on deciding certain order-theoretic properties, primarily the properties of being well-quasi-ordered (WQO) and of being atomic. In Chapter 1, we establish the background material required for the remainder of the thesis. This will include concepts from order theory, formal language theory, automata theory, and the theory of permutations. We also introduce various orderings on words, and the consecutive involvement ordering on permutations. In Chapter 2, we consider the prefix, suffix and factor orderings on words. For the prefix and suffix orderings, we give a characterisation of the regular languages which are WQO, and of those which are atomic. We then consider the factor ordering and show that the atomicity is decidable for finitely-based sets. We also give a new proof that WQO is decidable for finitely-based sets, which is a special case of a result of Atminas et al. In Chapters 3 and 4, we consider some general families of orderings on words. In Chapter 3 we consider orderings on words which are rational, meaning that they can be generated by transducers. We discuss the class of insertion relations introduced in a paper by the author, and introduce a generalisation. In Chapter 4, we consider three other variations of orderings on words. Throughout these chapters we prove various decidability results. In Chapter 5, we consider the consecutive involvement on permutations. We generalise our results for the factor ordering on words to show that WQO and atomicity are decidable. Through this investigation we answer some questions which have been asked (and remain open) for the involvement on permutations. 2019-12-03T00:00:00Z McDevitt, Matthew Substructure orderings are ubiquitous throughout combinatorics and all of mathematics. In this thesis we consider various orderings on words, as well as the consecutive involvement ordering on permutations. Throughout there will be a focus on deciding certain order-theoretic properties, primarily the properties of being well-quasi-ordered (WQO) and of being atomic. In Chapter 1, we establish the background material required for the remainder of the thesis. This will include concepts from order theory, formal language theory, automata theory, and the theory of permutations. We also introduce various orderings on words, and the consecutive involvement ordering on permutations. In Chapter 2, we consider the prefix, suffix and factor orderings on words. For the prefix and suffix orderings, we give a characterisation of the regular languages which are WQO, and of those which are atomic. We then consider the factor ordering and show that the atomicity is decidable for finitely-based sets. We also give a new proof that WQO is decidable for finitely-based sets, which is a special case of a result of Atminas et al. In Chapters 3 and 4, we consider some general families of orderings on words. In Chapter 3 we consider orderings on words which are rational, meaning that they can be generated by transducers. We discuss the class of insertion relations introduced in a paper by the author, and introduce a generalisation. In Chapter 4, we consider three other variations of orderings on words. Throughout these chapters we prove various decidability results. In Chapter 5, we consider the consecutive involvement on permutations. We generalise our results for the factor ordering on words to show that WQO and atomicity are decidable. Through this investigation we answer some questions which have been asked (and remain open) for the involvement on permutations. Assouad type dimensions and dimension spectra http://hdl.handle.net/10023/18157 In the first part of this thesis we introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r)ᵅ balls of radius r. To each 𝛩 ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/log r = 𝛩. The resulting 'dimension spectrum' (as a function of 𝛩) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding 'lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic and geometric properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension. We compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers. In the second part of this thesis, we study the Assouad dimension of sets of integers and deduce a weak solution to the Erdős-Turán conjecture. Let 𝐹 ⊂ ℕ. If $\sum_{n\in F}n^{-1}=\infty$ then 𝐹 "asymptotically" contains arbitrarily long arithmetic progressions. 2019-12-03T00:00:00Z Yu, Han In the first part of this thesis we introduce a new dimension spectrum motivated by the Assouad dimension; a familiar notion of dimension which, for a given metric space, returns the minimal exponent α ≥ 0 such that for any pair of scales 0 < r < R, any ball of radius R may be covered by a constant times (R/r)ᵅ balls of radius r. To each 𝛩 ∈ (0,1), we associate the appropriate analogue of the Assouad dimension with the restriction that the two scales r and R used in the definition satisfy log R/log r = 𝛩. The resulting 'dimension spectrum' (as a function of 𝛩) thus gives finer geometric information regarding the scaling structure of the space and, in some precise sense, interpolates between the upper box dimension and the Assouad dimension. This latter point is particularly useful because the spectrum is generally better behaved than the Assouad dimension. We also consider the corresponding 'lower spectrum', motivated by the lower dimension, which acts as a dual to the Assouad spectrum. We conduct a detailed study of these dimension spectra; including analytic and geometric properties. We also compute the spectra explicitly for some common examples of fractals including decreasing sequences with decreasing gaps and spirals with sub-exponential and monotonic winding. We also give several applications of our results, including: dimension distortion estimates under bi-Hölder maps for Assouad dimension. We compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of 'tail densities' defined for subsets of the integers. In the second part of this thesis, we study the Assouad dimension of sets of integers and deduce a weak solution to the Erdős-Turán conjecture. Let 𝐹 ⊂ ℕ. If $\sum_{n\in F}n^{-1}=\infty$ then 𝐹 "asymptotically" contains arbitrarily long arithmetic progressions. Semigroup congruences : computational techniques and theoretical applications http://hdl.handle.net/10023/17350 Computational semigroup theory is an area of research that is subject to growing interest. The development of semigroup algorithms allows for new theoretical results to be discovered, which in turn informs the creation of yet more algorithms. Groups have benefitted from this cycle since before the invention of electronic computers, and the popularity of computational group theory has resulted in a rich and detailed literature. Computational semigroup theory is a less developed field, but recent work has resulted in a variety of algorithms, and some important pieces of software such as the Semigroups package for GAP. Congruences are an important part of semigroup theory. A semigroup’s congruences determine its homomorphic images in a manner analogous to a group’s normal subgroups. Prior to the work described here, there existed few practical algorithms for computing with semigroup congruences. However, a number of results about alternative representations for congruences, as well as existing algorithms that can be borrowed from group theory, make congruences a fertile area for improvement. In this thesis, we first consider computational techniques that can be applied to the study of congruences, and then present some results that have been produced or precipitated by applying these techniques to interesting examples. After some preliminary theory, we present a new parallel approach to computing with congruences specified by generating pairs. We then consider alternative ways of representing a congruence, using intermediate objects such as linked triples. We also present an algorithm for computing the entire congruence lattice of a finite semigroup. In the second part of the thesis, we classify the congruences of several monoids of bipartitions, as well as the principal factors of several monoids of partial transformations. Finally, we consider how many congruences a finite semigroup can have, and examine those on semigroups with up to seven elements. 2019-06-25T00:00:00Z Torpey, Michael Computational semigroup theory is an area of research that is subject to growing interest. The development of semigroup algorithms allows for new theoretical results to be discovered, which in turn informs the creation of yet more algorithms. Groups have benefitted from this cycle since before the invention of electronic computers, and the popularity of computational group theory has resulted in a rich and detailed literature. Computational semigroup theory is a less developed field, but recent work has resulted in a variety of algorithms, and some important pieces of software such as the Semigroups package for GAP. Congruences are an important part of semigroup theory. A semigroup’s congruences determine its homomorphic images in a manner analogous to a group’s normal subgroups. Prior to the work described here, there existed few practical algorithms for computing with semigroup congruences. However, a number of results about alternative representations for congruences, as well as existing algorithms that can be borrowed from group theory, make congruences a fertile area for improvement. In this thesis, we first consider computational techniques that can be applied to the study of congruences, and then present some results that have been produced or precipitated by applying these techniques to interesting examples. After some preliminary theory, we present a new parallel approach to computing with congruences specified by generating pairs. We then consider alternative ways of representing a congruence, using intermediate objects such as linked triples. We also present an algorithm for computing the entire congruence lattice of a finite semigroup. In the second part of the thesis, we classify the congruences of several monoids of bipartitions, as well as the principal factors of several monoids of partial transformations. Finally, we consider how many congruences a finite semigroup can have, and examine those on semigroups with up to seven elements. Computational techniques in finite semigroup theory http://hdl.handle.net/10023/16521 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. 2019-06-25T00:00:00Z Wilson, Wilf A. 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. Some group presentations with few defining relations http://hdl.handle.net/10023/15964 We consider two classes of groups with two generators and three relations. One class has a similar presentation to groups considered in the paper by C.M. Campbell and R.M. Thomas, ‘On (2,n)-Groups related to Fibonacci Groups’, (Israel J. Math., 58), with one generator of order three instead of order two . We attempt to find the order of these groups and in one case find groups which have the alternating group A₅ as a subgroup of index equal to the order of the second generator of the group. Questions remain as to the order of some of the other groups. The second class has already been considered in the paper 'Some families of finite groups having two generators and two relations' by C.M. Campbell , H.S.M. Coxeter and E.F. Robertson, (Proc. R. Soc. London A. 357, 423-438 (1977)), in which a formula for the orders of these groups was found. We attempt to find simpler formulae based on recurrence relations for subclasses and write Maple programs to enable us to do this. We also find a formula, again based on recurrence relations, for an upper bound for the orders of the groups. 1990-01-01T00:00:00Z Gill, David Michael We consider two classes of groups with two generators and three relations. One class has a similar presentation to groups considered in the paper by C.M. Campbell and R.M. Thomas, ‘On (2,n)-Groups related to Fibonacci Groups’, (Israel J. Math., 58), with one generator of order three instead of order two . We attempt to find the order of these groups and in one case find groups which have the alternating group A₅ as a subgroup of index equal to the order of the second generator of the group. Questions remain as to the order of some of the other groups. The second class has already been considered in the paper 'Some families of finite groups having two generators and two relations' by C.M. Campbell , H.S.M. Coxeter and E.F. Robertson, (Proc. R. Soc. London A. 357, 423-438 (1977)), in which a formula for the orders of these groups was found. We attempt to find simpler formulae based on recurrence relations for subclasses and write Maple programs to enable us to do this. We also find a formula, again based on recurrence relations, for an upper bound for the orders of the groups. Decision problems in groups of homeomorphisms of Cantor space http://hdl.handle.net/10023/15885 The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$ and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and $G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of $F$ and $T$ were characterised in the seminal paper of Brin () and, later on, Brin and Guzman () investigate automorphisms of $T_{n, n-1}$ and $F_{n}$ for $n>2$. However, their techniques give no information about automorphisms of $G_{n,r}$. The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article , we show that automorphisms of $G_{n,r}$ are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition. In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article  of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on . We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure. Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h⁻¹ f (h)\phi$. This defines an equivalence relation on $G$ and $G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$. In the final chapter we show, using the description of $\aut{G_{n,r}}$, that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble. 2018-12-06T00:00:00Z Olukoya, Feyisayo The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$ and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and $G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of $F$ and $T$ were characterised in the seminal paper of Brin () and, later on, Brin and Guzman () investigate automorphisms of $T_{n, n-1}$ and $F_{n}$ for $n>2$. However, their techniques give no information about automorphisms of $G_{n,r}$. The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article , we show that automorphisms of $G_{n,r}$ are homeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition. In the rest of Chapter 2 and early sections of Chapter 3 we investigate the group $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article  of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the group of automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on . We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure. Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h⁻¹ f (h)\phi$. This defines an equivalence relation on $G$ and $G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$. In the final chapter we show, using the description of $\aut{G_{n,r}}$, that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble. On plausible counterexamples to Lehnert's conjecture http://hdl.handle.net/10023/15631 A group whose co-word problem is a context free language is called co𝐶𝐹 . Lehnert's conjecture states that a group 𝐺 is co𝐶𝐹 if and only if 𝐺 embeds as a finitely generated subgroup of R. Thompson's group V . In this thesis we explore a class of groups, Faug, proposed by Berns-Zieze, Fry, Gillings, Hoganson, and Mathews to contain potential counterexamples to Lehnert's conjecture. We create infinite and finite presentations for such groups and go on to prove that a certain subclass of 𝓕𝑎𝑢𝑔 consists of groups that do embed into 𝑉. By Anisimov a group has regular word problem if and only if it is finite. It is also known that a group 𝐺 is finite if and only if there exists an embedding of 𝐺 into 𝑉 such that its natural action on 𝕮₂:= {0, 1}[super]𝜔 is free on the whole space. We show that the class of groups with a context free word problem, the class of 𝐶𝐹 groups, is precisely the class of finitely generated demonstrable groups for 𝑉 . A demonstrable group for V is a group 𝐺 which is isomorphic to a subgroup in 𝑉 whose natural action on 𝕮₂ acts freely on an open subset. Thus our result extends the correspondence between language theoretic properties of groups and dynamical properties of subgroups of V . Additionally, our result also shows that the final condition of the four known closure properties of the class of co𝐶𝐹 groups also holds for the set of finitely generated subgroups of 𝑉. 2018-01-01T00:00:00Z Bennett, Daniel A group whose co-word problem is a context free language is called co𝐶𝐹 . Lehnert's conjecture states that a group 𝐺 is co𝐶𝐹 if and only if 𝐺 embeds as a finitely generated subgroup of R. Thompson's group V . In this thesis we explore a class of groups, Faug, proposed by Berns-Zieze, Fry, Gillings, Hoganson, and Mathews to contain potential counterexamples to Lehnert's conjecture. We create infinite and finite presentations for such groups and go on to prove that a certain subclass of 𝓕𝑎𝑢𝑔 consists of groups that do embed into 𝑉. By Anisimov a group has regular word problem if and only if it is finite. It is also known that a group 𝐺 is finite if and only if there exists an embedding of 𝐺 into 𝑉 such that its natural action on 𝕮₂:= {0, 1}[super]𝜔 is free on the whole space. We show that the class of groups with a context free word problem, the class of 𝐶𝐹 groups, is precisely the class of finitely generated demonstrable groups for 𝑉 . A demonstrable group for V is a group 𝐺 which is isomorphic to a subgroup in 𝑉 whose natural action on 𝕮₂ acts freely on an open subset. Thus our result extends the correspondence between language theoretic properties of groups and dynamical properties of subgroups of V . Additionally, our result also shows that the final condition of the four known closure properties of the class of co𝐶𝐹 groups also holds for the set of finitely generated subgroups of 𝑉. Commutativity and free products in Thompson's group V http://hdl.handle.net/10023/14652 We broaden the theory of dynamical interpretation, investigate the property of commutativity and explore the subject of subgroups forming free products in Thompson's group V. We expand Brin's terminology for a revealing pair to an any tree pair. We use it to analyse the dynamical behaviour of an arbitrary tree pair which cannot occur in a revealing pair. Hence, we design a series of algorithms generating Brin's revealing pair from any tree pair, by successively eliminating the undesirable structures. To detect patterns and transitioning between tree pairs, we introduce a new combinatorial object called the chains graph. A newly defined, unique and symmetrical type of a tree pair, called a balanced tree pair, stems from the use of the chains graphs. The main theorem of Bleak et al. in "Centralizers in the R. Thompson's Group V_n" states the necessary structure of the centraliser of an element of V. We provide a converse to this theorem, by proving that each of the predicted structures is realisable. Hence we obtain a complete classification of centralisers in V. We give an explicit construction of an element of V with prescribed centraliser. The underlying concept is to embed a Cayley graph of a finite group into the flow graph (introduced in Bleak et al.) of the desired element. To reflect the symmetry, we present the resulting element in terms of a balanced tree pair. The group V is conjectured to be a universal coCF group, which generates interest in studying its subgroups. We develop a better understanding of embeddings into V by providing a necessary and sufficient dynamical condition for two subgroups (not both torsion) to form a free product in V. For this, we use the properties, explored in Bleak and Salazar-Díaz "Free Products in Thompson's Group V", of sets of so--called important points, and the Ping-Pong action induced on them. 2018-06-26T00:00:00Z Bieniecka, Ewa We broaden the theory of dynamical interpretation, investigate the property of commutativity and explore the subject of subgroups forming free products in Thompson's group V. We expand Brin's terminology for a revealing pair to an any tree pair. We use it to analyse the dynamical behaviour of an arbitrary tree pair which cannot occur in a revealing pair. Hence, we design a series of algorithms generating Brin's revealing pair from any tree pair, by successively eliminating the undesirable structures. To detect patterns and transitioning between tree pairs, we introduce a new combinatorial object called the chains graph. A newly defined, unique and symmetrical type of a tree pair, called a balanced tree pair, stems from the use of the chains graphs. The main theorem of Bleak et al. in "Centralizers in the R. Thompson's Group V_n" states the necessary structure of the centraliser of an element of V. We provide a converse to this theorem, by proving that each of the predicted structures is realisable. Hence we obtain a complete classification of centralisers in V. We give an explicit construction of an element of V with prescribed centraliser. The underlying concept is to embed a Cayley graph of a finite group into the flow graph (introduced in Bleak et al.) of the desired element. To reflect the symmetry, we present the resulting element in terms of a balanced tree pair. The group V is conjectured to be a universal coCF group, which generates interest in studying its subgroups. We develop a better understanding of embeddings into V by providing a necessary and sufficient dynamical condition for two subgroups (not both torsion) to form a free product in V. For this, we use the properties, explored in Bleak and Salazar-Díaz "Free Products in Thompson's Group V", of sets of so--called important points, and the Ping-Pong action induced on them. Proof search issues in some non-classical logics http://hdl.handle.net/10023/13362 This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4. 1999-01-01T00:00:00Z Howe, Jacob M. This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4. The construction of finite soluble factor groups of finitely presented groups and its application http://hdl.handle.net/10023/12600 Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified. 1992-01-01T00:00:00Z Wegner, Alexander Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified. Counting subwords and other results related to the generalised star-height problem for regular languages http://hdl.handle.net/10023/12024 The Generalised Star-Height Problem is an open question in the field of formal language theory that concerns a measure of complexity on the class of regular languages; specifically, it asks whether or not there exists an algorithm to determine the generalised star-height of a given regular language. Rather surprisingly, it is not yet known whether there exists a regular language of generalised star-height greater than one. Motivated by a theorem of Thérien, we first take a combinatorial approach to the problem and consider the languages in which every word features a fixed contiguous subword an exact number of times. We show that these languages are all of generalised star-height zero. Similarly, we consider the languages in which every word features a fixed contiguous subword a prescribed number of times modulo a fixed number and show that these languages are all of generalised star-height at most one. Using these combinatorial results, we initiate work on identifying the generalised star-height of the languages that are recognised by finite semigroups. To do this, we establish the generalised star-height of languages recognised by Rees zero-matrix semigroups over nilpotent groups of classes zero and one before considering Rees zero-matrix semigroups over monogenic semigroups. Finally, we explore the generalised star-height of languages recognised by finite groups of a given order. We do this through the use of finite state automata and 'count arrows' to examine semidirect products of the form 𝐴 ⋊ ℤ[sub]𝑟, where 𝐴 is an abelian group and ℤ[sub]𝑟 is the cyclic group of order 𝑟. 2017-12-07T00:00:00Z Bourne, Thomas The Generalised Star-Height Problem is an open question in the field of formal language theory that concerns a measure of complexity on the class of regular languages; specifically, it asks whether or not there exists an algorithm to determine the generalised star-height of a given regular language. Rather surprisingly, it is not yet known whether there exists a regular language of generalised star-height greater than one. Motivated by a theorem of Thérien, we first take a combinatorial approach to the problem and consider the languages in which every word features a fixed contiguous subword an exact number of times. We show that these languages are all of generalised star-height zero. Similarly, we consider the languages in which every word features a fixed contiguous subword a prescribed number of times modulo a fixed number and show that these languages are all of generalised star-height at most one. Using these combinatorial results, we initiate work on identifying the generalised star-height of the languages that are recognised by finite semigroups. To do this, we establish the generalised star-height of languages recognised by Rees zero-matrix semigroups over nilpotent groups of classes zero and one before considering Rees zero-matrix semigroups over monogenic semigroups. Finally, we explore the generalised star-height of languages recognised by finite groups of a given order. We do this through the use of finite state automata and 'count arrows' to examine semidirect products of the form 𝐴 ⋊ ℤ[sub]𝑟, where 𝐴 is an abelian group and ℤ[sub]𝑟 is the cyclic group of order 𝑟. Fractal, group theoretic, and relational structures on Cantor space http://hdl.handle.net/10023/11370 Cantor space, the set of infinite words over a finite alphabet, is a type of metric space with a self-similar' structure. This thesis explores three areas concerning Cantor space with regard to fractal geometry, group theory, and topology. We find first results on the dimension of intersections of fractal sets within the Cantor space. More specifically, we examine the intersection of a subset E of the n-ary Cantor space, C[sub]n with the image of another subset Funder a random isometry. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of the Cantor space built from transducers. These groups can be seen as homeomorphisms that respect the self-similar and symmetric structure of C[sub]n, and are supergroups of the Higman-Thompson groups V[sub]n. We explore their isomorphism classes with our primary result being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is a semiregular subgroup of the symmetric group on n points. Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing these quotients by invariant relations as an inverse limit, before characterizing a specific class of fractals known as Sierpiński relatives as invariant factors. We then compare relations arising through edge replacement systems to invariant relations, detailing the conditions under which they are the same. 2016-01-01T00:00:00Z Donoven, Casey Ryall Cantor space, the set of infinite words over a finite alphabet, is a type of metric space with a self-similar' structure. This thesis explores three areas concerning Cantor space with regard to fractal geometry, group theory, and topology. We find first results on the dimension of intersections of fractal sets within the Cantor space. More specifically, we examine the intersection of a subset E of the n-ary Cantor space, C[sub]n with the image of another subset Funder a random isometry. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of the Cantor space built from transducers. These groups can be seen as homeomorphisms that respect the self-similar and symmetric structure of C[sub]n, and are supergroups of the Higman-Thompson groups V[sub]n. We explore their isomorphism classes with our primary result being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is a semiregular subgroup of the symmetric group on n points. Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing these quotients by invariant relations as an inverse limit, before characterizing a specific class of fractals known as Sierpiński relatives as invariant factors. We then compare relations arising through edge replacement systems to invariant relations, detailing the conditions under which they are the same. Constructing 2-generated subgroups of the group of homeomorphisms of Cantor space http://hdl.handle.net/10023/11362 We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, the group of homeomorphisms of Cantor space. In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]c is vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]c is closed under several natural constructions. In Chapter 4 we use a generalised notion of word and the tight completion construction from  to construct a family of subgroups of H[sub]c which generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples. 2017-01-01T00:00:00Z Hyde, James Thomas We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, the group of homeomorphisms of Cantor space. In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]c is vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]c is closed under several natural constructions. In Chapter 4 we use a generalised notion of word and the tight completion construction from  to construct a family of subgroups of H[sub]c which generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples. Generalized Bernstein polynomials and total positivity http://hdl.handle.net/10023/11183 "This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex... Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q for the class of convex functions. Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials." -- from the Abstract. 1999-01-01T00:00:00Z Oruç, Halil "This thesis submitted for Ph.D. degree deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex... Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q for the class of convex functions. Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials." -- from the Abstract. Flatness, extension and amalgamation in monoids, semigroups and rings http://hdl.handle.net/10023/11071 We begin our study of amalgamations by examining some ideas which are well-known for the category of R-modules. In particular we look at such notions as direct limits, pushouts, pullbacks, tensor products and flatness in the category of S-sets. Chapter II introduces the important concept of free extensions and uses this to describe the amalgamated free product. In Chapter III we define the extension property and the notion of purity. We show that many of the important notions in semigroup amalgams are intimately connected to these. In Section 2 we deduce that 'the extension property implies amalgamation' and more surprisingly that a semigroup U is an amalgamation base if and only if it has the extension property in every containing semigroup. Chapter IV revisits the idea of flatness and after some technical results we prove a result, similar to one for rings, on flat amalgams. In Chapter V we show that the results of Hall and Howie on perfect amalgams can be proved using the same techniques as those used in Chapters III and IV. We conclude the thesis with a look at the case of rings. We show that almost all of the results for semi group amalgams examined in the previous chapters, also hold for ring amalgams. 1986-01-01T00:00:00Z Renshaw, James Henry We begin our study of amalgamations by examining some ideas which are well-known for the category of R-modules. In particular we look at such notions as direct limits, pushouts, pullbacks, tensor products and flatness in the category of S-sets. Chapter II introduces the important concept of free extensions and uses this to describe the amalgamated free product. In Chapter III we define the extension property and the notion of purity. We show that many of the important notions in semigroup amalgams are intimately connected to these. In Section 2 we deduce that 'the extension property implies amalgamation' and more surprisingly that a semigroup U is an amalgamation base if and only if it has the extension property in every containing semigroup. Chapter IV revisits the idea of flatness and after some technical results we prove a result, similar to one for rings, on flat amalgams. In Chapter V we show that the results of Hall and Howie on perfect amalgams can be proved using the same techniques as those used in Chapters III and IV. We conclude the thesis with a look at the case of rings. We show that almost all of the results for semi group amalgams examined in the previous chapters, also hold for ring amalgams. Dimension theory of random self-similar and self-affine constructions http://hdl.handle.net/10023/11033 This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always maximal' in both measure theoretic and topological meanings. 2017-06-23T00:00:00Z Troscheit, Sascha This thesis is structured as follows. Chapter 1 introduces fractal sets before recalling basic mathematical concepts from dynamical systems, measure theory, dimension theory and probability theory. In Chapter 2 we give an overview of both deterministic and stochastic sets obtained from iterated function systems. We summarise classical results and set most of the basic notation. This is followed by the introduction of random graph directed systems in Chapter 3, based on the single authored paper [T1] to be published in Journal of Fractal Geometry. We prove that these attractors have equal Hausdorff and upper box-counting dimension irrespective of overlaps. It follows that the same holds for the classical models introduced in Chapter 2. This chapter also contains results about the Assouad dimensions for these random sets. Chapter 4 is based on the single authored paper [T2] and establishes the box-counting dimension for random box-like self-affine sets using some of the results and the notation developed in Chapter 3. We give some examples to illustrate the results. In Chapter 5 we consider the Hausdorff and packing measure of random attractors and show that for reasonable random systems the Hausdorff measure is zero almost surely. We further establish bounds on the gauge functions necessary to obtain positive or finite Hausdorff measure for random homogeneous systems. Chapter 6 is based on a joint article with J. M. Fraser and J.-J. Miao [FMT] to appear in Ergodic Theory and Dynamical Systems. It is chronologically the first and contains results that were extended in the paper on which Chapter 3 is based. However, we will give some simpler, alternative proofs in this section and crucially also find the Assouad dimension of some random self-affine carpets and show that the Assouad dimension is always maximal' in both measure theoretic and topological meanings. Restricted permutations, antichains, atomic classes and stack sorting http://hdl.handle.net/10023/11023 Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attracted the attention of some celebrated mathematicians including Paul Erdős and, due to its close links with sorting devices, Donald Knuth. We compare and contrast two presentations of closed classes that depend on the partial order of involvement: Basis or Avoidance Set, and Union of Atomic Classes. We examine how the basis is affected by a comprehensive list of closed class constructions and decompositions. The partial order of involvement contains infinite antichains. We develop the concept of a fundamental antichain. We compare the concept of 'fundamental' with other definitions of minimality for antichains, and compare fundamental permutation antichains with fundamental antichains in graph theory. The justification for investigating fundamental antichains is the nice patterns they produce. We forward the case for classifying the fundamental permutation antichains. Sorting devices have close links with closed classes. We consider two sorting devices, constructed from stacks in series, in detail. We give a comment on an enumerative conjecture by Ira Gessel. We demonstrate, with a remarkable example, that there exist two closed classes, equinumerous, one of which has a single basis element, the other infinitely many basis elements. We present this paper as a comprehensive analysis of the partial order of permutation involvement. We regard the main research contributions offered here to be the examples that demonstrate what is, and what is not, possible; although there are numerous structure results that do not fall under this category. We propose the classification of fundamental permutation antichains as one of the principal problems for closed classes today, and consider this as a problem whose solution will have wide significance for the study of partial orders, and mathematics as a whole. 2003-01-01T00:00:00Z Murphy, Maximilian M. Involvement is a partial order on all finite permutations, of infinite dimension and having subsets isomorphic to every countable partial order with finite descending chains. It has attracted the attention of some celebrated mathematicians including Paul Erdős and, due to its close links with sorting devices, Donald Knuth. We compare and contrast two presentations of closed classes that depend on the partial order of involvement: Basis or Avoidance Set, and Union of Atomic Classes. We examine how the basis is affected by a comprehensive list of closed class constructions and decompositions. The partial order of involvement contains infinite antichains. We develop the concept of a fundamental antichain. We compare the concept of 'fundamental' with other definitions of minimality for antichains, and compare fundamental permutation antichains with fundamental antichains in graph theory. The justification for investigating fundamental antichains is the nice patterns they produce. We forward the case for classifying the fundamental permutation antichains. Sorting devices have close links with closed classes. We consider two sorting devices, constructed from stacks in series, in detail. We give a comment on an enumerative conjecture by Ira Gessel. We demonstrate, with a remarkable example, that there exist two closed classes, equinumerous, one of which has a single basis element, the other infinitely many basis elements. We present this paper as a comprehensive analysis of the partial order of permutation involvement. We regard the main research contributions offered here to be the examples that demonstrate what is, and what is not, possible; although there are numerous structure results that do not fall under this category. We propose the classification of fundamental permutation antichains as one of the principal problems for closed classes today, and consider this as a problem whose solution will have wide significance for the study of partial orders, and mathematics as a whole. Multifractal zeta functions http://hdl.handle.net/10023/10637 Multifractals have during the past 20 − 25 years been the focus of enormous attention in the mathematical literature. Loosely speaking there are two main ingredients in multifractal analysis: the multifractal spectra and the Renyi dimensions. One of the main goals in multifractal analysis is to understand these two ingredients and their relationship with each other. Motivated by the powerful techniques provided by the use of the Artin-Mazur zeta-functions in number theory and the use of the Ruelle zeta-functions in dynamical systems, Lapidus and collaborators (see books by Lapidus & van Frankenhuysen [32, 33] and the references therein) have introduced and pioneered use of zeta-functions in fractal geometry. Inspired by this development, within the past 7−8 years several authors have paralleled this development by introducing zeta-functions into multifractal geometry. Our result inspired by this work will be given in section 2.2.2. There we introduce geometric multifractal zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. Results in that section are based on paper . Dynamical zeta-functions have been introduced and developed by Ruelle [63, 64] and others, (see, for example, the surveys and books [3, 54, 55] and the references therein). It has been a major challenge to introduce and develop a natural and meaningful theory of dynamical multifractal zeta-functions paralleling existing theory of dynamical zeta functions. In particular, in the setting of self-conformal constructions, Olsen  introduced a family of dynamical multifractal zeta-functions designed to provide precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. However, recently it has been recognised that while self-conformal constructions provide a useful and important framework for studying fractal and multifractal geometry, the more general notion of graph-directed self-conformal constructions provide a substantially more flexible and useful framework, see, for example,  for an elaboration of this. In recognition of this viewpoint, in section 2.3.11 we provide main definitions of the multifractal pressure and the multifractal dynamical zeta-functions and we state our main results. This section is based on paper . Setting we are working unifies various different multifractal spectra including fine multifractal spectra of self-conformal measures or Birkhoff averages of continuous function. It was introduced by Olsen in . In section 2.1 we propose answer to problem of defining Renyi spectra in more general settings and provide slight improvement of result regrading multifractal spectra in the case of Subshift of finite type. 2017-06-23T00:00:00Z Mijović, Vuksan Multifractals have during the past 20 − 25 years been the focus of enormous attention in the mathematical literature. Loosely speaking there are two main ingredients in multifractal analysis: the multifractal spectra and the Renyi dimensions. One of the main goals in multifractal analysis is to understand these two ingredients and their relationship with each other. Motivated by the powerful techniques provided by the use of the Artin-Mazur zeta-functions in number theory and the use of the Ruelle zeta-functions in dynamical systems, Lapidus and collaborators (see books by Lapidus & van Frankenhuysen [32, 33] and the references therein) have introduced and pioneered use of zeta-functions in fractal geometry. Inspired by this development, within the past 7−8 years several authors have paralleled this development by introducing zeta-functions into multifractal geometry. Our result inspired by this work will be given in section 2.2.2. There we introduce geometric multifractal zeta-functions providing precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. Results in that section are based on paper . Dynamical zeta-functions have been introduced and developed by Ruelle [63, 64] and others, (see, for example, the surveys and books [3, 54, 55] and the references therein). It has been a major challenge to introduce and develop a natural and meaningful theory of dynamical multifractal zeta-functions paralleling existing theory of dynamical zeta functions. In particular, in the setting of self-conformal constructions, Olsen  introduced a family of dynamical multifractal zeta-functions designed to provide precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. However, recently it has been recognised that while self-conformal constructions provide a useful and important framework for studying fractal and multifractal geometry, the more general notion of graph-directed self-conformal constructions provide a substantially more flexible and useful framework, see, for example,  for an elaboration of this. In recognition of this viewpoint, in section 2.3.11 we provide main definitions of the multifractal pressure and the multifractal dynamical zeta-functions and we state our main results. This section is based on paper . Setting we are working unifies various different multifractal spectra including fine multifractal spectra of self-conformal measures or Birkhoff averages of continuous function. It was introduced by Olsen in . In section 2.1 we propose answer to problem of defining Renyi spectra in more general settings and provide slight improvement of result regrading multifractal spectra in the case of Subshift of finite type. Generating "large" subgroups and subsemigroups http://hdl.handle.net/10023/9913 In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: (i) We classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; (ii) We investigate topological “largeness”, in particular subgroups which are finitely generated and dense; (iii) We investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion. 2016-01-01T00:00:00Z Jonušas, Julius In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: (i) We classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; (ii) We investigate topological “largeness”, in particular subgroups which are finitely generated and dense; (iii) We investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion. Synchronizing permutation groups and graph endomorphisms http://hdl.handle.net/10023/9912 The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms . The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from  is extended to semigroups of the form <G, T>, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8). 2016-01-01T00:00:00Z Schaefer, Artur The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms . The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from  is extended to semigroups of the form <G, T>, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8). Aspects of order and congruence relations on regular semigroups http://hdl.handle.net/10023/8926 On a regular semigroup S natural order relations have been defined by Nambooripad and by Lallement. Different characterisations and relationships between the Nambooripad order J, Lallement's order λ and a certain relation k are considered in Chapter I. It is shown that on a regular semigroup S the partial order J is left compatible if and only if S is locally R-unipotent. This condition in the case where S is orthodox is equivalent to saying that E(S) is a left seminormal band. It is also proved that λ is the least compatible partial order contained in J and that k = λ if and only if k is compatible and k if and only if J is compatible. A description of λ and J in the semigroups T(X) and PT(X) is presented. In Chapter II, it is proved that in an orthodox semigroup S the band of idempotents E(S) is left quasinormal if and only if there exists a local isomorphism from S onto an R-unipotent semigroup. It is shown that there exists a least R-unipotent congruence on any orthodox semigroup, generated by a certain left compatible equivalence R. This equivalence is a congruence if and only if E(S) is a right semiregular band. The last Chapter is particularly concerned with the description of R-unipotent congruences on a regular semigroup S by means of their kernels and traces. The lattice RC(S) of all R-unipotent congruences on a regular semigroup S is studied. A congruence≡ on the lattice RC(S) is considered and the greatest and the least element of each ≡-class are described. 1983-01-01T00:00:00Z Gomes, Gracinda Maria dos Santos On a regular semigroup S natural order relations have been defined by Nambooripad and by Lallement. Different characterisations and relationships between the Nambooripad order J, Lallement's order λ and a certain relation k are considered in Chapter I. It is shown that on a regular semigroup S the partial order J is left compatible if and only if S is locally R-unipotent. This condition in the case where S is orthodox is equivalent to saying that E(S) is a left seminormal band. It is also proved that λ is the least compatible partial order contained in J and that k = λ if and only if k is compatible and k if and only if J is compatible. A description of λ and J in the semigroups T(X) and PT(X) is presented. In Chapter II, it is proved that in an orthodox semigroup S the band of idempotents E(S) is left quasinormal if and only if there exists a local isomorphism from S onto an R-unipotent semigroup. It is shown that there exists a least R-unipotent congruence on any orthodox semigroup, generated by a certain left compatible equivalence R. This equivalence is a congruence if and only if E(S) is a right semiregular band. The last Chapter is particularly concerned with the description of R-unipotent congruences on a regular semigroup S by means of their kernels and traces. The lattice RC(S) of all R-unipotent congruences on a regular semigroup S is studied. A congruence≡ on the lattice RC(S) is considered and the greatest and the least element of each ≡-class are described. Graph automatic semigroups http://hdl.handle.net/10023/8645 In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FA-presentable semigroups. We consider the properties of graph automatic semigroups, showing that they are independent of the choice of generating set, have decidable word problem, and that if we have a graph automatic structure for a semigroup then we can find one with uniqueness. Semigroup constructions and their effect on graph automaticity are considered. We show that finitely generated direct products, free products, finitely generated Rees matrix semigroup constructions, zero unions, and ordinal sums all preserve unary graph automaticity, and examine when the converse also holds. We also demonstrate situations where semidirect products, Bruck-Reilly extensions, and semilattice constructions preserve graph automaticity, and consider the conditions we may impose on such constructions in order to ensure that graph automaticity is preserved. Unary graph automatic semigroups, that is semigroups which have graph automatic structures over a single letter alphabet, are also examined. We consider the form of an automaton recognising multiplication by generators in such a semigroup, and use this to demonstrate various properties of unary graph automatic semigroups. We show that infinite periodic semigroups are not unary graph automatic, and show that we may always find a uniform set of normal forms for a unary graph automatic semigroup. We also determine some necessary conditions for a semigroup to be unary graph automatic, and use this to provide examples of semigroups which are not unary graph automatic. Finally we consider semigroup constructions for unary graph automatic semigroups. We show that the free product of two semigroups is unary graph automatic if and only if both semigroups are trivial; that direct products do not always preserve unary graph automaticity; and that Bruck-Reilly extensions are never unary graph automatic. 2016-06-24T00:00:00Z Carey, Rachael Marie In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FA-presentable semigroups. We consider the properties of graph automatic semigroups, showing that they are independent of the choice of generating set, have decidable word problem, and that if we have a graph automatic structure for a semigroup then we can find one with uniqueness. Semigroup constructions and their effect on graph automaticity are considered. We show that finitely generated direct products, free products, finitely generated Rees matrix semigroup constructions, zero unions, and ordinal sums all preserve unary graph automaticity, and examine when the converse also holds. We also demonstrate situations where semidirect products, Bruck-Reilly extensions, and semilattice constructions preserve graph automaticity, and consider the conditions we may impose on such constructions in order to ensure that graph automaticity is preserved. Unary graph automatic semigroups, that is semigroups which have graph automatic structures over a single letter alphabet, are also examined. We consider the form of an automaton recognising multiplication by generators in such a semigroup, and use this to demonstrate various properties of unary graph automatic semigroups. We show that infinite periodic semigroups are not unary graph automatic, and show that we may always find a uniform set of normal forms for a unary graph automatic semigroup. We also determine some necessary conditions for a semigroup to be unary graph automatic, and use this to provide examples of semigroups which are not unary graph automatic. Finally we consider semigroup constructions for unary graph automatic semigroups. We show that the free product of two semigroups is unary graph automatic if and only if both semigroups are trivial; that direct products do not always preserve unary graph automaticity; and that Bruck-Reilly extensions are never unary graph automatic. Dimension and measure theory of self-similar structures with no separation condition http://hdl.handle.net/10023/7854 We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0. 2015-11-30T00:00:00Z Farkas, Ábel We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0. On generators, relations and D-simplicity of direct products, Byleen extensions, and other semigroup constructions http://hdl.handle.net/10023/7629 In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable. The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank. 2015-11-30T00:00:00Z Baynes, Samuel In this thesis we study two different topics, both in the context of semigroup constructions. The first is the investigation of an embedding problem, specifically the problem of whether it is possible to embed any given finitely presentable semigroup into a D-simple finitely presentable semigroup. We consider some well-known semigroup constructions, investigating their properties to determine whether they might prove useful for finding a solution to our problem. We carry out a more detailed study into a more complicated semigroup construction, the Byleen extension, which has been used to solve several other embedding problems. We prove several results regarding the structure of this extension, finding necessary and sufficient conditions for an extension to be D-simple and a very strong necessary condition for an extension to be finitely presentable. The second topic covered in this thesis is relative rank, specifically the sequence obtained by taking the rank of incremental direct powers of a given semigroup modulo the diagonal subsemigroup. We investigate the relative rank sequences of infinite Cartesian products of groups and of semigroups. We characterise all semigroups for which the relative rank sequence of an infinite Cartesian product is finite, and show that if the sequence is finite then it is bounded above by a logarithmic function. We will find sufficient conditions for the relative rank sequence of an infinite Cartesian product to be logarithmic, and sufficient conditions for it to be constant. Chapter 4 ends with the introduction of a new topic, relative presentability, which follows naturally from the topic of relative rank. The maximal subgroups of the classical groups in dimension 13, 14 and 15 http://hdl.handle.net/10023/7067 One might easily argue that the Classification of Finite Simple Groups is one of the most important theorems of group theory. Given that any finite group can be deconstructed into its simple composition factors, it is of great importance to have a detailed knowledge of the structure of finite simple groups. One of the classes of finite groups that appear in the classification theorem are the simple classical groups, which are matrix groups preserving some form. This thesis will shed some new light on almost simple classical groups in dimension 13, 14 and 15. In particular we will determine their maximal subgroups. We will build on the results by Bray, Holt, and Roney-Dougal who calculated the maximal subgroups of all almost simple finite classical groups in dimension less than 12. Furthermore, Aschbacher proved that the maximal subgroups of almost simple classical groups lie in nine classes. The maximal subgroups in the first eight classes, i.e. the subgroups of geometric type, were determined by Kleidman and Liebeck for dimension greater than 13. Therefore this thesis concentrates on the ninth class of Aschbacher’s Theorem. This class roughly consists of subgroups which are almost simple modulo scalars and do not preserve a geometric structure. As our final result we will give tables containing all maximal subgroups of almost simple classical groups in dimension 13, 14 and 15. 2015-11-30T00:00:00Z Schröder, Anna Katharina One might easily argue that the Classification of Finite Simple Groups is one of the most important theorems of group theory. Given that any finite group can be deconstructed into its simple composition factors, it is of great importance to have a detailed knowledge of the structure of finite simple groups. One of the classes of finite groups that appear in the classification theorem are the simple classical groups, which are matrix groups preserving some form. This thesis will shed some new light on almost simple classical groups in dimension 13, 14 and 15. In particular we will determine their maximal subgroups. We will build on the results by Bray, Holt, and Roney-Dougal who calculated the maximal subgroups of all almost simple finite classical groups in dimension less than 12. Furthermore, Aschbacher proved that the maximal subgroups of almost simple classical groups lie in nine classes. The maximal subgroups in the first eight classes, i.e. the subgroups of geometric type, were determined by Kleidman and Liebeck for dimension greater than 13. Therefore this thesis concentrates on the ninth class of Aschbacher’s Theorem. This class roughly consists of subgroups which are almost simple modulo scalars and do not preserve a geometric structure. As our final result we will give tables containing all maximal subgroups of almost simple classical groups in dimension 13, 14 and 15. Dots and lines : geometric semigroup theory and finite presentability http://hdl.handle.net/10023/6923 Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these. 2015-06-26T00:00:00Z Awang, Jennifer S. Geometric semigroup theory means different things to different people, but it is agreed that it involves associating a geometric structure to a semigroup and deducing properties of the semigroup based on that structure. One such property is finite presentability. In geometric group theory, the geometric structure of choice is the Cayley graph of the group. It is known that in group theory finite presentability is an invariant under quasi-isometry of Cayley graphs. We choose to associate a metric space to a semigroup based on a Cayley graph of that semigroup. This metric space is constructed by removing directions, multiple edges and loops from the Cayley graph. We call this a skeleton of the semigroup. We show that finite presentability of certain types of direct products, completely (0-)simple, and Clifford semigroups is preserved under isomorphism of skeletons. A major tool employed in this is the Švarc-Milnor Lemma. We present an example that shows that in general, finite presentability is not an invariant property under isomorphism of skeletons of semigroups, and in fact is not an invariant property under quasi-isometry of Cayley graphs for semigroups. We give several skeletons and describe fully the semigroups that can be associated to these. Cayley automaton semigroups http://hdl.handle.net/10023/6558 Let 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. 2015-06-26T00:00:00Z McLeman, Alexander Lewis Andrew Let 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. Dimension theory and fractal constructions based on self-affine carpets http://hdl.handle.net/10023/3869 The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from affine' to bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'. 2013-11-29T00:00:00Z Fraser, Jonathan M. The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from affine' to bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'. Finiteness conditions for unions of semigroups http://hdl.handle.net/10023/3687 In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem. 2013-06-28T00:00:00Z Abu-Ghazalh, Nabilah Hani In this thesis we prove the following: The semigroup which is a disjoint union of two or three copies of a group is a Clifford semigroup, Rees matrix semigroup or a combination between a Rees matrix semigroup and a group. Furthermore, the semigroup which is a disjoint union of finitely many copies of a finitely presented (residually finite) group is finitely presented (residually finite) semigroup. The constructions of the semigroup which is a disjoint union of two copies of the free monogenic semigroup are parallel to the constructions of the semigroup which is a disjoint union of two copies of a group, i.e. such a semigroup is Clifford (strong semilattice of groups) or Rees matrix semigroup. However, the semigroup which is a disjoint union of three copies of the free monogenic semigroup is not just a strong semillatice of semigroups, Rees matrix semigroup or combination between a Rees matrix semigroup and a semigroup, but there are two more semigroups which do not arise from the constructions of the semigroup which is a disjoint union of three copies of a group. We also classify semigroups which are disjoint unions of two or three copies of the free monogenic semigroup. There are three types of semigroups which are unions of two copies of the free monogenic semigroup and nine types of semigroups which are unions of three copies of the free monogenic semigroup. For each type of such semigroups we exhibit a presentation defining semigroups of this type. The semigroup which is a disjoint union of finitely many copies of the free monogenic semigroup is finitely presented, residually finite, hopfian, has soluble word problem and has soluble subsemigroup membership problem. Adventures in applying iteration lemmas http://hdl.handle.net/10023/3671 The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set. The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton. We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata. We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation. We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently. A central tool used in the derivation of structural results are so-called iteration lemmas. We define a hierarchy of the considered classes of semigroups and connect our original results with previous research. 2013-06-28T00:00:00Z Pfeiffer, Markus Johannes The word problem of a finitely generated group is commonly defined to be a formal language over a finite generating set. The class of finite groups has been characterised as the class of finitely generated groups that have word problem decidable by a finite state automaton. We give a natural generalisation of the notion of word problem from finitely generated groups to finitely generated semigroups by considering relations of strings. We characterise the class of finite semigroups by the class of finitely generated semigroups whose word problem is decidable by finite state automata. We then examine the class of semigroups with word problem decidable by asynchronous two tape finite state automata. Algebraic properties of semigroups in this class are considered, towards an algebraic characterisation. We take the next natural step to further extend the classes of semigroups under consideration to semigroups that have word problem decidable by a finite collection of asynchronous automata working independently. A central tool used in the derivation of structural results are so-called iteration lemmas. We define a hierarchy of the considered classes of semigroups and connect our original results with previous research. Ends of semigroups http://hdl.handle.net/10023/3590 The aim of this thesis is to understand the algebraic structure of a semigroup by studying the geometric properties of its Cayley graph. We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopfs Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups. We classify all semigroups with one end and make use of this classification to prove various finiteness properties for semigroups with one end. We also consider the ends of digraphs with certain algebraic properties. We prove that two quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph. 2013-01-01T00:00:00Z Craik, Simon The aim of this thesis is to understand the algebraic structure of a semigroup by studying the geometric properties of its Cayley graph. We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopfs Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups. We classify all semigroups with one end and make use of this classification to prove various finiteness properties for semigroups with one end. We also consider the ends of digraphs with certain algebraic properties. We prove that two quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph. Random generation and chief length of finite groups http://hdl.handle.net/10023/3578 Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian simple group G with d randomly chosen elements, and extends this idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability of generating an almost simple group G by d randomly chosen elements, given that they project onto a generating set of G/Soc(G). In particular we show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90, with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10 except for 30 almost simple groups G, and we specify this list and provide exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150. In Part II we consider a related notion. Given a probability ε, we wish to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G), the minimal number of generators needed to generate G. We obtain bounds on the chief length of permutation groups in terms of the degree n, and bounds on the chief length of completely reducible matrix groups in terms of the dimension and field size. Combining these with existing bounds on d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely reducible matrix groups. 2013-01-01T00:00:00Z Menezes, Nina E. Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian simple group G with d randomly chosen elements, and extends this idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability of generating an almost simple group G by d randomly chosen elements, given that they project onto a generating set of G/Soc(G). In particular we show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90, with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10 except for 30 almost simple groups G, and we specify this list and provide exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150. In Part II we consider a related notion. Given a probability ε, we wish to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G), the minimal number of generators needed to generate G. We obtain bounds on the chief length of permutation groups in terms of the degree n, and bounds on the chief length of completely reducible matrix groups in terms of the dimension and field size. Combining these with existing bounds on d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely reducible matrix groups. Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures http://hdl.handle.net/10023/3358 Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H-classes. 2012-11-30T00:00:00Z McPhee, Jillian Dawn Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H-classes. Topics in combinatorial semigroup theory http://hdl.handle.net/10023/3226 In this thesis we discuss various topics from Combinatorial Semigroup Theory: automaton semigroups; finiteness conditions and their preservation under certain semigroup theoretic notions of index; Markov semigroups; word-hyperbolic semigroups; decision problems for finitely presented and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups. We investigate how various semigroup finiteness conditions, linked to the notion of ideal, are preserved under finite Rees and Green indices. We obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption, hopficity is preserved by finite Rees index extensions. Still, there is an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness. This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate how the property of being Markov is preserved under finite Rees and Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and one-relator monoids. 2012-11-30T00:00:00Z Maltcev, Victor In this thesis we discuss various topics from Combinatorial Semigroup Theory: automaton semigroups; finiteness conditions and their preservation under certain semigroup theoretic notions of index; Markov semigroups; word-hyperbolic semigroups; decision problems for finitely presented and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman’s property. We prove that an automaton semigroup generated by a Cayley machine of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups. We investigate how various semigroup finiteness conditions, linked to the notion of ideal, are preserved under finite Rees and Green indices. We obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption, hopficity is preserved by finite Rees index extensions. Still, there is an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness. This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate how the property of being Markov is preserved under finite Rees and Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and one-relator monoids. Presentations and efficiency of semigroups http://hdl.handle.net/10023/2843 In this thesis we consider in detail the following two problems for semigroups: (i) When are semigroups finitely generated and presented? (ii) Which families of semigroups can be efficiently presented? We also consider some other finiteness conditions for semigroups, homology of semigroups and wreath product of groups. In Chapter 2 we investigate finite presentability and some other finiteness conditions for the O-direct union of semigroups with zero. In Chapter 3 we investigate finite generation and presentability of Rees matrix semigroups over semigroups. We find necessary and sufficient conditions for finite generation and presentability. In Chapter 4 we investigate some other finiteness conditions for Rees matrix semigroups. In Chapter 5 we consider groups as semigroups and investigate their semigroup efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups that are not groups. We first give examples of efficient and inefficient "proper" semigroups by computing their homology and finding their minimal presentations. In Chapter 7 we compute the second homology of finite simple semigroups and find a "small" presentation for them. If that "small" presentation has a special relation, we prove that finite simple semigroups are efficient. Finally, in Chapter 8, we investigate the efficiency of wreath products of finite groups as groups and as semigroups. We give more examples of efficient groups and inefficient groups. 1998-01-01T00:00:00Z Ayik, Hayrullah In this thesis we consider in detail the following two problems for semigroups: (i) When are semigroups finitely generated and presented? (ii) Which families of semigroups can be efficiently presented? We also consider some other finiteness conditions for semigroups, homology of semigroups and wreath product of groups. In Chapter 2 we investigate finite presentability and some other finiteness conditions for the O-direct union of semigroups with zero. In Chapter 3 we investigate finite generation and presentability of Rees matrix semigroups over semigroups. We find necessary and sufficient conditions for finite generation and presentability. In Chapter 4 we investigate some other finiteness conditions for Rees matrix semigroups. In Chapter 5 we consider groups as semigroups and investigate their semigroup efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups that are not groups. We first give examples of efficient and inefficient "proper" semigroups by computing their homology and finding their minimal presentations. In Chapter 7 we compute the second homology of finite simple semigroups and find a "small" presentation for them. If that "small" presentation has a special relation, we prove that finite simple semigroups are efficient. Finally, in Chapter 8, we investigate the efficiency of wreath products of finite groups as groups and as semigroups. We give more examples of efficient groups and inefficient groups. Semigroups of order-decreasing transformations http://hdl.handle.net/10023/2834 Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated. 1992-01-01T00:00:00Z Umar, Abdullahi Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial one-to-one) transformations of X. In this Thesis the study of order-increasing full (partial, partial one-to-one) transformations has been reduced to that of order-decreasing full (partial, partial one-to-one) transformations and the study of order-decreasing partial transformations to that of order-decreasing full transformations for both the finite and infinite cases. For the finite order-decreasing full (partial one-to-one) transformation semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden (1990) concerning products of idempotents (quasi-idempotents), and concerning combinatorial and rank properties. By contrast with the semigroups of order-preserving transformations and the full transformation semigroup, the semigroups of orderdecreasing full (partial one-to-one) transformations and their Rees quotient semigroups are not regular. They are, however, abundant (type A) semigroups in the sense of Fountain (1982,1979). An explicit characterisation of the minimum semilattice congruence on the finite semigroups of order-decreasing transformations and their Rees quotient semigroups is obtained. If X is an infinite chain then the semigroup S of order-decreasing full transformations need not be abundant. A necessary and sufficient condition on X is obtained for S to be abundant. By contrast, for every chain X the semigroup of order-decreasing partial one-to-one transformations is type A. The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing full (partial one-to-one) transformations have been investigated. Semigroup presentations http://hdl.handle.net/10023/2821 In this thesis we consider in detail the following two fundamental problems for semigroup presentations: 1. Given a semigroup find a presentation defining it. 2. Given a presentation describe the semigroup defined by it. We also establish two links between these two approaches: semigroup constructions and computational methods. After an introduction to semigroup presentations in Chapter 3, in Chapters 4 and 5 we consider the first of the two approaches. The semigroups we examine in these two chapters include completely O-simple semigroups, transformation semigroups, matrix semigroups and various endomorphism semigroups. In Chapter 6 we find presentations for the following semi group constructions: wreath product, Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids, Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in more detail presentations for subsemigroups in Chapters 7 and 10, where we prove a number of Reidemeister-Schreier type results for semigroups. In Chapter 9 we examine the connection between the semi group and the group defined by the same presentation. The general results from Chapters 6, 7, 9 and 10 are applied in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci semigroups, semigroups defined by Coxeter type presentations and one relator products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter enumeration procedure and introduce three modifications of this procedure. 1995-01-01T00:00:00Z Ruškuc, Nik In this thesis we consider in detail the following two fundamental problems for semigroup presentations: 1. Given a semigroup find a presentation defining it. 2. Given a presentation describe the semigroup defined by it. We also establish two links between these two approaches: semigroup constructions and computational methods. After an introduction to semigroup presentations in Chapter 3, in Chapters 4 and 5 we consider the first of the two approaches. The semigroups we examine in these two chapters include completely O-simple semigroups, transformation semigroups, matrix semigroups and various endomorphism semigroups. In Chapter 6 we find presentations for the following semi group constructions: wreath product, Bruck-Reilly extension, Schiitzenberger product, strong semilattices of monoids, Rees matrix semigroups, ideal extensions and subsemigroups. We investigate in more detail presentations for subsemigroups in Chapters 7 and 10, where we prove a number of Reidemeister-Schreier type results for semigroups. In Chapter 9 we examine the connection between the semi group and the group defined by the same presentation. The general results from Chapters 6, 7, 9 and 10 are applied in Chapters 8, 11, 12 and 13 to subsemigroups of free semigroups, Fibonacci semigroups, semigroups defined by Coxeter type presentations and one relator products of cyclic groups. Finally, in Chapter 14 we describe the Todd-Coxeter enumeration procedure and introduce three modifications of this procedure. Topics in computational group theory : primitive permutation groups and matrix group normalisers http://hdl.handle.net/10023/2561 Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm. 2011-11-01T00:00:00Z Coutts, Hannah Jane Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm. Generation problems for finite groups http://hdl.handle.net/10023/2529 It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property. 2011-11-30T00:00:00Z McDougall-Bagnall, Jonathan M. It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property. Directed graph iterated function systems http://hdl.handle.net/10023/2109 This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections of the components of Fᵤ, is strictly less than the multifractal q Hausdorff dimension with respect to μᵤ of Fᵤ. 2011-11-30T00:00:00Z Boore, Graeme C. This thesis concerns an active research area within fractal geometry. In the first part, in Chapters 2 and 3, for directed graph iterated function systems (IFSs) defined on ℝ, we prove that a class of 2-vertex directed graph IFSs have attractors that cannot be the attractors of standard (1-vertex directed graph) IFSs, with or without separation conditions. We also calculate their exact Hausdorff measure. Thus we are able to identify a new class of attractors for which the exact Hausdorff measure is known. We give a constructive algorithm for calculating the set of gap lengths of any attractor as a finite union of cosets of finitely generated semigroups of positive real numbers. The generators of these semigroups are contracting similarity ratios of simple cycles in the directed graph. The algorithm works for any IFS defined on ℝ with no limit on the number of vertices in the directed graph, provided a separation condition holds. The second part, in Chapter 4, applies to directed graph IFSs defined on ℝⁿ . We obtain an explicit calculable value for the power law behaviour as r → 0⁺ , of the qth packing moment of μᵤ, the self-similar measure at a vertex u, for the non-lattice case, with a corresponding limit for the lattice case. We do this (i) for any q ∈ ℝ if the strong separation condition holds, (ii) for q ≥ 0 if the weaker open set condition holds and a specified non-negative matrix associated with the system is irreducible. In the non-lattice case this enables the rate of convergence of the packing L[superscript(q)]-spectrum of μᵤ to be determined. We also show, for (ii) but allowing q ∈ ℝ, that the upper multifractal q box-dimension with respect to μᵤ, of the set consisting of all the intersections of the components of Fᵤ, is strictly less than the multifractal q Hausdorff dimension with respect to μᵤ of Fᵤ. A commutative noncommutative fractal geometry http://hdl.handle.net/10023/1710 In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory. 2010-01-01T00:00:00Z Samuel, Anthony In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}}^{{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory. The lesser names : the teachers of the Edinburgh Mathematical Society and other aspects of Scottish mathematics, 1867–1946 http://hdl.handle.net/10023/1700 The Edinburgh Mathematical Society started out in 1883 as a society with a large proportion of teachers. Today, the member base is mainly academical and there are only a few teachers left. This thesis explores how and when this change came about, and discusses what this meant for the Society. It argues that the exit of the teachers is related to the rising standard of mathematics, but even more to a change in the Society’s printing policy in the 1920s, that turned the Society’s Proceedings into a pure research publication and led to the death of the ‘teacher journal’, the Mathematical Notes. The thesis also argues that this change, drastic as it may seem, does not represent a change in the Society’s nature. For this aim, the role of the teachers within the Society has been studied and compared to that of the academics, from 1883 to 1946. The mathematical contribution of the teachers to the Proceedings is studied in some detail, in particular the papers by John Watt Butters. A paper in the Mathematical Notes by A. C. Aitken on the Bell numbers is considered in connection with a series of letters on the same topic from 1938–39. These letters, written by Aitken, Sir D’Arcy Thompson, another EMS member, and the Cambridge mathematician G. T. Bennett, explores the relation between the three and gives valuable insight into the status of the Notes. Finally, the role of the first women in the Society is studied. The first woman joined without any official university education, but had received the necessary mathematical background from her studies under the Edinburgh Association for the University Education of Women. The final chapter is largely an assessment of this Association’s mathematical classes. 2011-06-22T00:00:00Z Hartveit, Marit The Edinburgh Mathematical Society started out in 1883 as a society with a large proportion of teachers. Today, the member base is mainly academical and there are only a few teachers left. This thesis explores how and when this change came about, and discusses what this meant for the Society. It argues that the exit of the teachers is related to the rising standard of mathematics, but even more to a change in the Society’s printing policy in the 1920s, that turned the Society’s Proceedings into a pure research publication and led to the death of the ‘teacher journal’, the Mathematical Notes. The thesis also argues that this change, drastic as it may seem, does not represent a change in the Society’s nature. For this aim, the role of the teachers within the Society has been studied and compared to that of the academics, from 1883 to 1946. The mathematical contribution of the teachers to the Proceedings is studied in some detail, in particular the papers by John Watt Butters. A paper in the Mathematical Notes by A. C. Aitken on the Bell numbers is considered in connection with a series of letters on the same topic from 1938–39. These letters, written by Aitken, Sir D’Arcy Thompson, another EMS member, and the Cambridge mathematician G. T. Bennett, explores the relation between the three and gives valuable insight into the status of the Notes. Finally, the role of the first women in the Society is studied. The first woman joined without any official university education, but had received the necessary mathematical background from her studies under the Edinburgh Association for the University Education of Women. The final chapter is largely an assessment of this Association’s mathematical classes. Stable and multistable processes and localisability http://hdl.handle.net/10023/948 We first review recent work on stable and multistable random processes and their localisability. Then most of the thesis concerns a new approach to these topics based on characteristic functions. Our aim is to construct processes on R, which are α(x)-multistable, where the stability index α(x) varies with x. To do this we first use characteristic functions to define α(x)-multistable random integrals and measures and examine their properties. We show that an α(x)-multistable random measure may be obtained as the limit of a sequence of measures made up of α-stable random measures restricted to small intervals with α constant on each interval. We then use the multistable random integrals to define multistable random processes on R and study the localisability of these processes. Thus we find conditions that ensure that a process locally ‘looks like’ a given stochastic process under enlargement and appropriate scaling. We give many examples of multistable random processes and examine their local forms. Finally, we examine the dimensions of graphs of α-stable random functions defined by series with α-stable random variables as coefficients. 2010-06-23T00:00:00Z Liu, Lining We first review recent work on stable and multistable random processes and their localisability. Then most of the thesis concerns a new approach to these topics based on characteristic functions. Our aim is to construct processes on R, which are α(x)-multistable, where the stability index α(x) varies with x. To do this we first use characteristic functions to define α(x)-multistable random integrals and measures and examine their properties. We show that an α(x)-multistable random measure may be obtained as the limit of a sequence of measures made up of α-stable random measures restricted to small intervals with α constant on each interval. We then use the multistable random integrals to define multistable random processes on R and study the localisability of these processes. Thus we find conditions that ensure that a process locally ‘looks like’ a given stochastic process under enlargement and appropriate scaling. We give many examples of multistable random processes and examine their local forms. Finally, we examine the dimensions of graphs of α-stable random functions defined by series with α-stable random variables as coefficients. Classification and enumeration of finite semigroups http://hdl.handle.net/10023/945 The 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. 2010-06-23T00:00:00Z Distler, Andreas The 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. Generating uncountable transformation semigroups http://hdl.handle.net/10023/867 We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions. (i) Is every countable subset F of S also a subset of a ﬁnitely generated subsemigroup of S? If so, what is the least number n such that for every countable subset F of S there exist n elements of S that generate a subsemigroup of S containing F as a subset. (ii) Given a subset U of S, what is the least cardinality of a subset A of S such that the union of A and U is a generating set for S? (iii) Deﬁne a preorder relation ≤ on the subsets of S as follows. For subsets V and W of S write V ≤ W if there exists a countable subset C of S such that V is contained in the semigroup generated by the union of W and C. Given a subset U of S, where does U lie in the preorder ≤ on subsets of S? Semigroups S for which we answer question (i) include: the semigroups of the injec- tive functions and the surjective functions on a countably inﬁnite set; the semigroups of the increasing functions, the Lebesgue measurable functions, and the differentiable functions on the closed unit interval [0, 1]; and the endomorphism semigroup of the random graph. We investigate questions (ii) and (iii) in the case where S is the semigroup Ω[superscript Ω] of all functions on a countably inﬁnite set Ω. Subsets U of Ω[superscript Ω] under consideration are semigroups of Lipschitz functions on Ω with respect to discrete metrics on Ω and semigroups of endomorphisms of binary relations on Ω such as graphs or preorders. 2009-01-01T00:00:00Z Péresse, Yann We consider naturally occurring, uncountable transformation semigroups S and investigate the following three questions. (i) Is every countable subset F of S also a subset of a ﬁnitely generated subsemigroup of S? If so, what is the least number n such that for every countable subset F of S there exist n elements of S that generate a subsemigroup of S containing F as a subset. (ii) Given a subset U of S, what is the least cardinality of a subset A of S such that the union of A and U is a generating set for S? (iii) Deﬁne a preorder relation ≤ on the subsets of S as follows. For subsets V and W of S write V ≤ W if there exists a countable subset C of S such that V is contained in the semigroup generated by the union of W and C. Given a subset U of S, where does U lie in the preorder ≤ on subsets of S? Semigroups S for which we answer question (i) include: the semigroups of the injec- tive functions and the surjective functions on a countably inﬁnite set; the semigroups of the increasing functions, the Lebesgue measurable functions, and the differentiable functions on the closed unit interval [0, 1]; and the endomorphism semigroup of the random graph. We investigate questions (ii) and (iii) in the case where S is the semigroup Ω[superscript Ω] of all functions on a countably inﬁnite set Ω. Subsets U of Ω[superscript Ω] under consideration are semigroups of Lipschitz functions on Ω with respect to discrete metrics on Ω and semigroups of endomorphisms of binary relations on Ω such as graphs or preorders. The geometry of self-affine fractals http://hdl.handle.net/10023/838 In this thesis we study the dimension theory of self-affine sets. We begin by introducing a number of notions from fractal geometry, in particular, dimensions, measure properties and iterated functions systems. We give a review of existing work on self-affine sets. We then develop a variety of new results on self-affine sets and their dimensional properties. This work falls into three parts: Firstly, we look at the dimension formulae for a class of self-affine sets generated by upper triangular matrices. In this case, we simplify the affine dimension formula into equations only involving the diagonal elements of the matrices. Secondly, since the Hausdorff dimensions of self-affine sets depend not only on the linear parts of the contractions but also on the translation parameters, we obtain an upper bound for the dimensions of exceptional sets, that is, the set of parameters such that the Hausdorff dimension of the attractor is smaller than the affine dimension. Thirdly, we investigate dimensions of a class of random self-affine sets, aiming to extend the ‘almost sure’ formula for random self-similar sets to random self-affine sets. 2008-01-01T00:00:00Z Miao, Jun Jie In this thesis we study the dimension theory of self-affine sets. We begin by introducing a number of notions from fractal geometry, in particular, dimensions, measure properties and iterated functions systems. We give a review of existing work on self-affine sets. We then develop a variety of new results on self-affine sets and their dimensional properties. This work falls into three parts: Firstly, we look at the dimension formulae for a class of self-affine sets generated by upper triangular matrices. In this case, we simplify the affine dimension formula into equations only involving the diagonal elements of the matrices. Secondly, since the Hausdorff dimensions of self-affine sets depend not only on the linear parts of the contractions but also on the translation parameters, we obtain an upper bound for the dimensions of exceptional sets, that is, the set of parameters such that the Hausdorff dimension of the attractor is smaller than the affine dimension. Thirdly, we investigate dimensions of a class of random self-affine sets, aiming to extend the ‘almost sure’ formula for random self-similar sets to random self-affine sets. Intersection problems in combinatorics http://hdl.handle.net/10023/765 With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets. We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits: if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size . The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain. The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects. 2009-11-30T00:00:00Z Brunk, Fiona With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets. We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits: if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size . The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain. The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects. Resonances for graph directed Markov systems, and geometry of infinitely generated dynamical systems http://hdl.handle.net/10023/719 In the first part of this thesis we transfer a result of Guillopé et al. concerning the number of zeros of the Selberg zeta function for convex cocompact Schottky groups to the setting of certain types of graph directed Markov systems (GDMS). For these systems the zeta function will be a type of Ruelle zeta function. We show that for a finitely generated primitive conformal GDMS S, which satisfies the strong separation condition (SSC) and the nestedness condition (NC), we have for each c>0 that the following holds, for each w \in\$C$ with Re(w)>-c, |\Im(w)|>1 and for all k \in\$N$ sufficiently large: log | zeta(w) | <<e^{delta(S).log(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k^{delta(S)}. Here, Q(k)\subset\%C$denotes a certain box of height k, and delta(S) refers to the Hausdorff dimension of the limit set of S. In the second part of this thesis we show that in any dimension m \in\$N$there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). Furthermore, we derive various relations between the exponents of convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems of finitely generated GDMSs. Finally, we apply our results to Kleinian groups and generalise a result of Patterson by showing that in any dimension m \in\$N$there are Kleinian groups for which the Hausdorff dimension of their uniformly radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). 2009-06-24T00:00:00Z Hille, Martial R. In the first part of this thesis we transfer a result of Guillopé et al. concerning the number of zeros of the Selberg zeta function for convex cocompact Schottky groups to the setting of certain types of graph directed Markov systems (GDMS). For these systems the zeta function will be a type of Ruelle zeta function. We show that for a finitely generated primitive conformal GDMS S, which satisfies the strong separation condition (SSC) and the nestedness condition (NC), we have for each c>0 that the following holds, for each w \in\$C$with Re(w)>-c, |\Im(w)|>1 and for all k \in\$N$sufficiently large: log | zeta(w) | <<e^{delta(S).log(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k^{delta(S)}. Here, Q(k)\subset\%C$ denotes a certain box of height k, and delta(S) refers to the Hausdorff dimension of the limit set of S. In the second part of this thesis we show that in any dimension m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). Furthermore, we derive various relations between the exponents of convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems of finitely generated GDMSs. Finally, we apply our results to Kleinian groups and generalise a result of Patterson by showing that in any dimension m \in\$N$ there are Kleinian groups for which the Hausdorff dimension of their uniformly radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). Inhomogeneous self-similar sets and measures http://hdl.handle.net/10023/682 The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L^{q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L^{q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures. 2008-01-01T00:00:00Z Snigireva, Nina The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous self-similar sets and measures. In particular, we show that these sets and measures are natural generalizations of the well known self-similar sets and measures. We then investigate the structure of these sets and measures. In the second chapter we study various fractal dimensions (Hausdorff, packing and box dimensions) of inhomogeneous self-similar sets and compare our results with the well-known results for (ordinary) self-similar sets. In the third chapter we investigate the L^{q} spectra and the Renyi dimensions of inhomogeneous self-similar measures and prove that new multifractal phenomena, not exhibited by (ordinary) self-similar measures, appear in the inhomogeneous case. Namely, we show that inhomogeneous self-similar measures may have phase transitions which is in sharp contrast to the behaviour of the L^{q} spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra of inhomogeneous self-similar measures. We show that the multifractal spectra of inhomogeneous self-similar measures may be non-concave which is again in sharp contrast to the behaviour of the multifractal spectra of (ordinary) self-similar measures satisfying the Open Set Condition. Then we present a number of applications of our results. Many of them are related to the notoriously difficult problem of computing (or simply obtaining non-trivial bounds) for the multifractal spectra of self-similar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain non-trivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of self-similar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of inhomogeneous self-similar measures and again we present a number of applications of our results, in particular to non-linear self-similar measures. Simplicity in relational structures and its application to permutation classes http://hdl.handle.net/10023/431 The simple relational structures form the units, or atoms, upon which all other relational structures are constructed by means of the substitution decomposition. This decomposition appears to have first been introduced in 1953 in a talk by FraÃ¯ssÃ©, though it did not appear in an article until a paper by Gallai in 1967. It has subsequently been frequently rediscovered from a wide variety of perspectives, ranging from game theory to combinatorial optimization. Of all the relational structures - a set which also includes graphs, tournaments and posets - permutations are receiving ever increasing amounts of attention. A simple permutation is one that maps every nontrivial contiguous set of indices to a set of indices that is never contiguous. Simple permutations and intervals of permutations are important in biomathematics, while permutation classes - downsets under the pattern containment order - arise naturally in settings ranging from sorting to algebraic geometry. We begin by studying simple permutations themselves, though always aim to establish this theory within the broader context of relational structures. We first develop the technology of "pin sequences", and prove that every sufficiently long simple permutation must contain either a long horizontal or parallel alternation, or a long pin sequence. This gives rise to a simpler unavoidable substructures result, namely that every sufficiently long simple permutation contains a long alternation or oscillation. ErdÅ s, Fried, Hajnal and Milner showed in 1972 that every tournament could be extended to a simple tournament by adding at most two additional points. We prove analogous results for permutations, graphs, and posets, noting that in these three cases we may need to extend a structure by adding (n+1)/2 points in the case of permutations and posets, and logâ (n+1) points in the graph case. The importance of simple permutations in permutation classes has been well established in recent years. We extend this knowledge in a variety of ways, first by showing that, in a permutation class containing only finitely many simple permutations, every subset defined by properties belonging to a finite "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time. Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based. 2007-11-30T00:00:00Z Brignall, Robert The simple relational structures form the units, or atoms, upon which all other relational structures are constructed by means of the substitution decomposition. This decomposition appears to have first been introduced in 1953 in a talk by FraÃ¯ssÃ©, though it did not appear in an article until a paper by Gallai in 1967. It has subsequently been frequently rediscovered from a wide variety of perspectives, ranging from game theory to combinatorial optimization. Of all the relational structures - a set which also includes graphs, tournaments and posets - permutations are receiving ever increasing amounts of attention. A simple permutation is one that maps every nontrivial contiguous set of indices to a set of indices that is never contiguous. Simple permutations and intervals of permutations are important in biomathematics, while permutation classes - downsets under the pattern containment order - arise naturally in settings ranging from sorting to algebraic geometry. We begin by studying simple permutations themselves, though always aim to establish this theory within the broader context of relational structures. We first develop the technology of "pin sequences", and prove that every sufficiently long simple permutation must contain either a long horizontal or parallel alternation, or a long pin sequence. This gives rise to a simpler unavoidable substructures result, namely that every sufficiently long simple permutation contains a long alternation or oscillation. ErdÅ s, Fried, Hajnal and Milner showed in 1972 that every tournament could be extended to a simple tournament by adding at most two additional points. We prove analogous results for permutations, graphs, and posets, noting that in these three cases we may need to extend a structure by adding (n+1)/2 points in the case of permutations and posets, and logâ (n+1) points in the graph case. The importance of simple permutations in permutation classes has been well established in recent years. We extend this knowledge in a variety of ways, first by showing that, in a permutation class containing only finitely many simple permutations, every subset defined by properties belonging to a finite "query-complete set" is enumerated by an algebraic generating function. Such properties include being an even or alternating permutation, or avoiding generalised (blocked or barred) permutations. We further indicate that membership of a permutation class containing only finitely many simple permutations can be computed in linear time. Using the decomposition of simple permutations, we establish, by representing pin sequences as a language over an eight-letter alphabet, that it is decidable if a permutation class given by a finite basis contains only finitely many simple permutations. We also discuss possible approaches to the same question for other relational structures, in particular the difficulties that arise for graphs. The pin sequence technology provides a further result relating to the wreath product of two permutation classes, namely that C â D is finitely based whenever D does not admit arbitrarily long pin sequences. As a partial converse, we also exhibit a number of explicit examples of wreath products that are not finitely based. On permutation classes defined by token passing networks, gridding matrices and pictures : three flavours of involvement http://hdl.handle.net/10023/237 The study of pattern classes is the study of the involvement order on finite permutations. This order can be traced back to the work of Knuth. In recent years the area has attracted the attention of many combinatoralists and there have been many structural and enumerative developments. We consider permutations classes defined in three different ways and demonstrate that asking the same fixed questions in each case motivates a different view of involvement. Token passing networks encourage us to consider permutations as sequences of integers; grid classes encourage us to consider them as point sets; picture classes, which are developed for the first time in this thesis, encourage a purely geometrical approach. As we journey through each area we present several new results. We begin by studying the basic definitions of a permutation. This is followed by a discussion of the questions one would wish to ask of permutation classes. We concentrate on four particular areas: partial well order, finite basis, atomicity and enumeration. Our third chapter asks these questions of token passing networks; we also develop the concept of completeness and show that it is decidable whether or not a particular network is complete. Next we move onto grid classes, our analysis using generic sets yields an algorithm for determining when a grid class is atomic; we also present a new and elegant proof which demonstrates that certain grid classes are partially well ordered. The final chapter comprises the development and analysis of picture classes. We completely classify and enumerate those permutations which can be drawn from a circle, those which can be drawn from an X and those which can be drawn from some convex polygon. We exhibit the first uncountable set of closed classes to be found in a natural setting; each class is drawn from three parallel lines. We present a permutation version of the famous happy ending' problem of Erdös and Szekeres. We conclude with a discussion of permutation classes in higher dimensional space. 2007-06-19T00:00:00Z Waton, Stephen D. The study of pattern classes is the study of the involvement order on finite permutations. This order can be traced back to the work of Knuth. In recent years the area has attracted the attention of many combinatoralists and there have been many structural and enumerative developments. We consider permutations classes defined in three different ways and demonstrate that asking the same fixed questions in each case motivates a different view of involvement. Token passing networks encourage us to consider permutations as sequences of integers; grid classes encourage us to consider them as point sets; picture classes, which are developed for the first time in this thesis, encourage a purely geometrical approach. As we journey through each area we present several new results. We begin by studying the basic definitions of a permutation. This is followed by a discussion of the questions one would wish to ask of permutation classes. We concentrate on four particular areas: partial well order, finite basis, atomicity and enumeration. Our third chapter asks these questions of token passing networks; we also develop the concept of completeness and show that it is decidable whether or not a particular network is complete. Next we move onto grid classes, our analysis using generic sets yields an algorithm for determining when a grid class is atomic; we also present a new and elegant proof which demonstrates that certain grid classes are partially well ordered. The final chapter comprises the development and analysis of picture classes. We completely classify and enumerate those permutations which can be drawn from a circle, those which can be drawn from an X and those which can be drawn from some convex polygon. We exhibit the first uncountable set of closed classes to be found in a natural setting; each class is drawn from three parallel lines. We present a permutation version of the famous happy ending' problem of Erdös and Szekeres. We conclude with a discussion of permutation classes in higher dimensional space.