Pure Mathematics Theses
http://hdl.handle.net/10023/99
20160630T12:11:10Z

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 Runipotent. 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 Runipotent semigroup. It is
shown that there exists a least Runipotent 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
Runipotent congruences on a regular semigroup S by means of their
kernels and traces. The lattice RC(S) of all Runipotent 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.
19830101T00:00:00Z
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 Runipotent. 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 Runipotent semigroup. It is
shown that there exists a least Runipotent 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
Runipotent congruences on a regular semigroup S by means of their
kernels and traces. The lattice RC(S) of all Runipotent 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 FApresentable 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, BruckReilly 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 BruckReilly extensions are never unary graph automatic.
20160624T00:00:00Z
Carey, Rachael Marie
In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FApresentable 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, BruckReilly 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 BruckReilly extensions are never unary graph automatic.

Dimension and measure theory of selfsimilar structures with no separation condition
http://hdl.handle.net/10023/7854
We introduce methods to cope with selfsimilar sets when we do not assume any separation condition. For a selfsimilar set K ⊆ ℝᵈ we establish a similarity dimensionlike 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 nonoverlapping selfsimilar sets to overlapping selfsimilar sets.
We investigate how the Hausdorff dimension and measure of a selfsimilar 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 selfsimilar 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 selfsimilar or graph directed selfsimilar 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 `selfsimilar'. 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 δapproximate packing premeasure coincide for sufficiently small δ > 0.
20151130T00:00:00Z
Farkas, Ábel
We introduce methods to cope with selfsimilar sets when we do not assume any separation condition. For a selfsimilar set K ⊆ ℝᵈ we establish a similarity dimensionlike 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 nonoverlapping selfsimilar sets to overlapping selfsimilar sets.
We investigate how the Hausdorff dimension and measure of a selfsimilar 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 selfsimilar 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 selfsimilar or graph directed selfsimilar 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 `selfsimilar'. 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 δapproximate packing premeasure coincide for sufficiently small δ > 0.

On generators, relations and Dsimplicity 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 Dsimple finitely presentable semigroup. We consider some wellknown 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 Dsimple 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.
20151130T00:00:00Z
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 Dsimple finitely presentable semigroup. We consider some wellknown 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 Dsimple 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 RoneyDougal 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.
20151130T00:00:00Z
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 RoneyDougal 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 quasiisometry 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 ŠvarcMilnor 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 quasiisometry of Cayley graphs for semigroups.
We give several skeletons and describe fully the semigroups that can be associated to these.
20150626T00:00:00Z
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 quasiisometry 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 ŠvarcMilnor 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 quasiisometry 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 nontrivial 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 leftregular 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.
20150626T00:00:00Z
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 nontrivial 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 leftregular 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 selfaffine carpets
http://hdl.handle.net/10023/3869
The aim of this thesis is to develop the dimension theory of selfaffine carpets in several directions. Selfaffine carpets are an important class of planar selfaffine 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 wellunderstood world of selfsimilar sets and the far from understood world of general selfaffine 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 selfsimilar 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 selfaffine 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 selfaffine carpets, which we call boxlike selfaffine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on selfaffine 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 selfaffine 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 quasiconformal 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 46 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1variable attractors, with the aim of developing the dimension theory of selfaffine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous selfsimilar 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 selfaffine setting and, in Chapter 5, investigate the dimensions of inhomogeneous selfaffine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of selfsimilar 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 selfaffine 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 `biLipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random selfsimilar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random selfaffine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
20131129T00:00:00Z
Fraser, Jonathan M.
The aim of this thesis is to develop the dimension theory of selfaffine carpets in several directions. Selfaffine carpets are an important class of planar selfaffine 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 wellunderstood world of selfsimilar sets and the far from understood world of general selfaffine 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 selfsimilar 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 selfaffine 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 selfaffine carpets, which we call boxlike selfaffine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on selfaffine 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 selfaffine 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 quasiconformal 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 46 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1variable attractors, with the aim of developing the dimension theory of selfaffine carpets in these directions.
In order to put our work into context, in Chapter 4 we consider inhomogeneous selfsimilar 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 selfaffine setting and, in Chapter 5, investigate the dimensions of inhomogeneous selfaffine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of selfsimilar 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 selfaffine 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 `biLipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random selfsimilar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random selfaffine 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.
20130628T00:00:00Z
AbuGhazalh, 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 socalled iteration lemmas.
We define a hierarchy of the considered classes of semigroups and connect our original results with previous research.
20130628T00:00:00Z
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 socalled 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 quasiisometric 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.
20130101T00:00:00Z
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 quasiisometric 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 2generated 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.
20130101T00:00:00Z
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 2generated 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 Hclasses, 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 alephnaught] 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 alephnaught] 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 alephnaught] regular Dclasses of End(Ω) and when Ω = R,D,B there exist 2[superscript alephnaught] Jclasses of End(Ω). Additionally we show
that if Ω = R,D then all regular Dclasses contain 2[superscript alephnaught] group Hclasses. On the other hand, we show that when
Ω = B,Q there exist regular Dclasses
which contain countably many group Hclasses.
20121130T00:00:00Z
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 Hclasses, 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 alephnaught] 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 alephnaught] 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 alephnaught] regular Dclasses of End(Ω) and when Ω = R,D,B there exist 2[superscript alephnaught] Jclasses of End(Ω). Additionally we show
that if Ω = R,D then all regular Dclasses contain 2[superscript alephnaught] group Hclasses. On the other hand, we show that when
Ω = B,Q there exist regular Dclasses
which contain countably many group Hclasses.

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; wordhyperbolic semigroups; decision problems for finitely presented
and onerelator 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 nonhopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent contextfree monadic rewriting systems are wordhyperbolic, and provide an example of such a monoid, which does not admit a wordhyperbolic 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
onerelator monoids.
20121130T00:00:00Z
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; wordhyperbolic semigroups; decision problems for finitely presented
and onerelator 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 nonhopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent contextfree monadic rewriting systems are wordhyperbolic, and provide an example of such a monoid, which does not admit a wordhyperbolic 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
onerelator 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 Odirect 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.
19980101T00:00:00Z
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 Odirect 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 orderdecreasing transformations
http://hdl.handle.net/10023/2834
Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial onetoone) transformations of X. In this
Thesis the study of orderincreasing full (partial, partial onetoone) transformations
has been reduced to that of orderdecreasing full (partial, partial onetoone)
transformations and the study of orderdecreasing partial transformations to that of
orderdecreasing full transformations for both the finite and infinite cases.
For the finite orderdecreasing full (partial onetoone) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasiidempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of orderpreserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial onetoone) 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 orderdecreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of orderdecreasing 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
orderdecreasing partial onetoone transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial onetoone) transformations have been investigated.
19920101T00:00:00Z
Umar, Abdullahi
Let X be a totally ordered set and consider the semigroups of orderdecreasing (increasing) full (partial, partial onetoone) transformations of X. In this
Thesis the study of orderincreasing full (partial, partial onetoone) transformations
has been reduced to that of orderdecreasing full (partial, partial onetoone)
transformations and the study of orderdecreasing partial transformations to that of
orderdecreasing full transformations for both the finite and infinite cases.
For the finite orderdecreasing full (partial onetoone) transformation
semigroups, we obtain results analogous to Howie (1971) and Howie and McFadden
(1990) concerning products of idempotents (quasiidempotents), and concerning
combinatorial and rank properties. By contrast with the semigroups of orderpreserving
transformations and the full transformation semigroup, the semigroups of orderdecreasing
full (partial onetoone) 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 orderdecreasing transformations and their Rees
quotient semigroups is obtained.
If X is an infinite chain then the semigroup S of orderdecreasing 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
orderdecreasing partial onetoone transformations is type A.
The ranks of the nilpotent subsemigroups of the finite semigroups of orderdecreasing
full (partial onetoone) 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 Osimple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
BruckReilly 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 ReidemeisterSchreier 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 ToddCoxeter
enumeration procedure and introduce three modifications of this procedure.
19950101T00:00:00Z
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 Osimple semigroups, transformation semigroups,
matrix semigroups and various endomorphism semigroups. In Chapter 6
we find presentations for the following semi group constructions: wreath product,
BruckReilly 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 ReidemeisterSchreier 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 ToddCoxeter
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'NanScott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'NanScott class. For the nonaffine 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.
20111101T00:00:00Z
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'NanScott Theorem
and Aschbacher's theorem. Tables of the groups G are given for each O'NanScott class. For the nonaffine 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 pgroup 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 primepower order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
20111130T00:00:00Z
McDougallBagnall, Jonathan M.
It can be deduced from the Burnside Basis Theorem that if G is a finite pgroup 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 primepower 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 2vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1vertex 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 selfsimilar measure at a vertex u, for the nonlattice 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 nonnegative
matrix associated with the system is irreducible.
In the nonlattice 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 boxdimension 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.
20111130T00:00:00Z
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 2vertex directed graph IFSs have attractors
that cannot be the attractors of standard (1vertex 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 selfsimilar measure at a vertex u, for the nonlattice 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 nonnegative
matrix associated with the system is irreducible.
In the nonlattice 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 boxdimension 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 nonempty 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 boxcounting dimension can be recovered. We show that for a selfsimilar 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 selfsimilar 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 AntonescuIvan and Christensen, we construct a family of (1, +)summable spectral triples for a onesided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the PerronFrobeniusRuelle operator, whose potential function is nonarithemetic and Hölder continuous. We show that the Connes' pseudometric, 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.
20100101T00:00:00Z
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 nonempty 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 boxcounting dimension can be recovered. We show that for a selfsimilar 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 selfsimilar 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 AntonescuIvan and Christensen, we construct a family of (1, +)summable spectral triples for a onesided topologically exact subshift of finite type (∑{{A}}^{{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the PerronFrobeniusRuelle operator, whose potential function is nonarithemetic and Hölder continuous. We show that the Connes' pseudometric, 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.
20110622T00:00:00Z
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 αstable 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 αstable random functions
defined by series with αstable random variables as coefficients.
20100623T00:00:00Z
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 αstable 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 αstable random functions
defined by series with αstable 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 antiisomorphism 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 antiisomorphic
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 antiisomorphism 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.
20100623T00:00:00Z
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 antiisomorphism 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 antiisomorphic
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 antiisomorphism 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.
20090101T00:00:00Z
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 selfaffine fractals
http://hdl.handle.net/10023/838
In this thesis we study the dimension theory of selfaffine 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 selfaffine sets. We then develop a variety of new results on selfaffine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of selfaffine 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 selfaffine 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 selfaffine sets, aiming
to extend the ‘almost sure’ formula for random selfsimilar sets to random selfaffine
sets.
20080101T00:00:00Z
Miao, Jun Jie
In this thesis we study the dimension theory of selfaffine 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 selfaffine sets. We then develop a variety of new results on selfaffine
sets and their dimensional properties.
This work falls into three parts:
Firstly, we look at the dimension formulae for a class of selfaffine 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 selfaffine 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 selfaffine sets, aiming
to extend the ‘almost sure’ formula for random selfsimilar sets to random selfaffine
sets.

Intersection problems in combinatorics
http://hdl.handle.net/10023/765
With the publication of the famous ErdősKoRado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is tintersecting if any two of its elements mutually tintersect, 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 1intersecting 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 socalled fixfamilies, consisting of all injections which map some fixed set of t points to the same image points, are the only tintersecting injection families of maximal size. By way of contrast, fixing the differences kt and nk while increasing k leads to optimal families which are equivalent to one of the socalled saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=_ (kt)/2 _. Furthermore we demonstrate that, among injection families with tintersecting and leftcompressed 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.
20091130T00:00:00Z
Brunk, Fiona
With the publication of the famous ErdősKoRado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is tintersecting if any two of its elements mutually tintersect, 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 1intersecting 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 socalled fixfamilies, consisting of all injections which map some fixed set of t points to the same image points, are the only tintersecting injection families of maximal size. By way of contrast, fixing the differences kt and nk while increasing k leads to optimal families which are equivalent to one of the socalled saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=_ (kt)/2 _. Furthermore we demonstrate that, among injection families with tintersecting and leftcompressed 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(Imw)} 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).
20090624T00:00:00Z
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(Imw)} 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 selfsimilar sets and measures
http://hdl.handle.net/10023/682
The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous selfsimilar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known selfsimilar 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 selfsimilar sets and compare our results with the wellknown results for (ordinary)
selfsimilar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous selfsimilar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) selfsimilar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous selfsimilar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) selfsimilar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous selfsimilar measures. We show that
the multifractal spectra
of
inhomogeneous selfsimilar
measures
may be nonconcave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) selfsimilar
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 nontrivial bounds) for the multifractal spectra of selfsimilar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain nontrivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of selfsimilar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous selfsimilar measures and again we present a
number of applications of our results, in particular to nonlinear
selfsimilar measures.
20080101T00:00:00Z
Snigireva, Nina
The thesis consists of four main chapters. The first chapter includes an introduction to inhomogeneous selfsimilar sets and
measures. In particular, we show that these sets and measures
are natural generalizations of the well known selfsimilar 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 selfsimilar sets and compare our results with the wellknown results for (ordinary)
selfsimilar sets. In the third chapter we investigate the L^{q}
spectra and the Renyi dimensions of inhomogeneous selfsimilar
measures and prove that new multifractal phenomena, not exhibited by (ordinary) selfsimilar measures, appear in the inhomogeneous case.
Namely, we show that inhomogeneous selfsimilar measures may
have phase transitions which is in sharp contrast to the
behaviour of the
L^{q} spectra
of (ordinary) selfsimilar
measures satisfying the Open Set Condition. Then we study the significantly more difficult problem of computing the multifractal spectra
of inhomogeneous selfsimilar measures. We show that
the multifractal spectra
of
inhomogeneous selfsimilar
measures
may be nonconcave which is again in sharp contrast to the
behaviour of the
multifractal spectra
of (ordinary) selfsimilar
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 nontrivial bounds) for the multifractal spectra of selfsimilar measures not satisfying the Open Set Condition. More precisely, we will show that our results provide a systematic approach to obtain nontrivial bounds (and in some cases even exact values) for the multifractal spectra of several large and interesting classes of selfsimilar measures not satisfying the Open Set Condition. In the fourth chapter we investigate the asymptotic behaviour of the Fourier transforms of
inhomogeneous selfsimilar measures and again we present a
number of applications of our results, in particular to nonlinear
selfsimilar 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 "querycomplete 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 eightletter 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.
20071130T00:00:00Z
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 "querycomplete 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 eightletter 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.
20070619T00:00:00Z
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.