In this thesis we discuss various topics from Combinatorial Semigroup Theory: automaton semigroups; finiteness conditions and their preservation under certain semigroup theoretic notions of index; Markov semigroups; word-hyperbolic semigroups; decision problems for finitely presented and one-relator monoids. First, in order to show that general ideas from Combinatorial Semigroup Theory can apply to uncountable semigroups, at the beginning of the thesis we discuss semigroups with Bergman's property. We prove that an automaton semigroup generated by a Cayley machine of a finite semigroup S is itself finite if and only if S is aperiodic, which yields a new characterisation of finite aperiodic monoids. Using this, we derive some further results about Cayley automaton semigroups. We investigate how various semigroup finiteness conditions, linked to the notion of ideal, are preserved under finite Rees and Green indices. We obtain a surprising result that J = D is preserved by supersemigroups of finite Green index, but it is not preserved by subsemigroups of finite Rees index even in the finitely generated case. We also consider the question of preservation of hopficity for finite Rees index. We prove that in general hopficity is preserved neither by finite Rees index subsemigroups, nor by finite Rees index extensions. However, under finite generation assumption, hopficity is preserved by finite Rees index extensions. Still, there is an example of a finitely generated hopfian semigroup with a non-hopfian subsemigroup of finite Rees index. We prove also that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic, and provide an example of such a monoid, which does not admit a word-hyperbolic structure with uniqueness. This answers in the negative a question of Duncan & Gilman. We initiate in this thesis a study of Markov semigroups. We investigate how the property of being Markov is preserved under finite Rees and Green indices. For various semigroup properties P we examine whether P , ¬P are Markov properties, and whether P is decidable for finitely presented and one-relator monoids.

The aim of this thesis is to understand the algebraic structure of a semigroup by studying the geometric properties of its Cayley graph. We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopfs Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups. We classify all semigroups with one end and make use of this classification to prove various finiteness properties for semigroups with one end. We also consider the ends of digraphs with certain algebraic properties. We prove that two quasi-isometric digraphs have isomorphic end sets. We also prove that vertex transitive digraphs have 1, 2 or infinitely many ends and construct a topology that reflects the properties of the ends of a digraph.

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Zappa-Szép products arise when a semigroup has the property that every element has a unique decomposition as a product of elements from two given subsemigroups. They may also be constructed from actions of two semigroups on one another satisfying axioms first formulated by G. Zappa.

In this thesis we study differentiability of C0-semigroups. In 1968 Pazy gave a characterization of differentiable C0-semigroups in terms of the generator. We begin by proving an alternative characterization of differentiable C0-semigroups. We then study bounded perturbations of the generators of differentiable Co-semigroups.

The topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasi-adequate, concordant, abundant, restriction, Ehresmann and weakly abundant semigroups. A semigroup $S$ with subset of idempotents U is weakly U-abundant if every $\art_U$-class and every $\elt_U$-class contains an idempotent of U, where $\art_U$ and $\elt_U$ are relations extending the well known Green's relations $\ar$ and $\el$. We assume throughout that our semigroups satisfy a condition known as the Congruence Condition (C). We take several approaches to weakly $U$-abundant semigroups. Our first results describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly $U$-abundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectangular bands of monoids. The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly $B$-orthodox semigroups, where $B$ is a band. We note that if $B$ is a semilattice then a weakly $B$-orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly $B$-orthodox semigroup $S$ as a spined product of a fundamental weakly $\overline{B}$-orthodox semigroup $S_B$ (depending only on $B$) and $S/\gamma_B$, where $\overline{B}$ is isomorphic to $B$ and $\gamma_B$ is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that $B$ is a normal band, or $S$ is weakly $B$-superabundant, we find a closed form $\delta_B$ for $\gamma_B$, which simplifies our result to a straightforward form. For the above to work smoothly in the case $S$ is weakly $B$-superabundant, we need to find a canonical fundamental weakly $B$-superabundant subsemigroup of $S_B$. This we do, and give the corresponding answers in the case of the Hall semigroup $W_B$ and a number of intervening semigroups. We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a regular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad's work to {\em weakly $U$-regular} semigroups, that is, weakly $U$-abundant semigroups with (C) and $U$ generating a regular subsemigroup whose set of idempotents is $U$. As an intervening step we consider weakly $B$-orthodox semigroups in this light. We take two approaches. The first is via a new construction of an inductive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids. We show that the category of weakly $B$-orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is influenced by that of Nambooripad, however, there are significant differences in strategy, the first being the introduction of generalised categories and the second being that it is more convenient to consider (generalised) categories equipped with pre-orders, rather than with partial orders. Our work may be regarded as extending a result of Lawson for Ehresmann semigroups. We also examine the trace of a weakly $B$-orthodox semigroup, which is a primitive weakly $B$-orthodox semigroup. We then take a more `traditional' approach to weakly $B$-orthodox semigroups via band categories and weakly orthodox categories over a band, equipped with two pre-orders. We show that the category of weakly $B$-orthodox semigroups is equivalent to the category of weakly orthodox categories over bands. To do so we must substantially adjust Armstrong's method for concordant semigroups. Finally, we consider the most general case of weakly $U$-regular semigroups. Following Nambooripad's theorem, which establishes a correspondence between algebraic structures (inverse semigroups) and ordered structures (inductive group-oids), we build a correspondence between the category of weakly $U$-regular semigroups and the category of weakly regular categories over regular biordered sets, equipped with two pre-orders.

Chapter 1 covers some basic notions and results from Algebraic Topology such as CW-complexes, homotopy and homology groups of a space in general and cellular homology for CW-complexes in particular. Also we give some basic ideas from abstract reduction systems and some supporting material such as several order relations on a set and the Knuth-Bendix completion procedure. There are only two original results of the author in this chapter, Theorem 1.4.5 and Theorem 1.7.3. The material related to Topology and Homological Algebra can be found in [12], [21], [40], [62], [82], [91] and [92]. The material related to reduction systems can be found in [5] and [11]. The original work of the author is included in Chapters 2, 3 and 4 apart from Section 3.2 which contains general notions from Category Theory, Section 3.5.2 which contains an account of the work in [67] and Section 4.1 which contains some basics from Combinatorial Semigroup Theory. The results of Section 4.2 are part of [83] which is accepted for publication in the International Journal of Algebra and Computation. The material related to Category Theory can be found in [59], [64], [66], [67], [74], [75], [76], [82] and [93]. The material related to Semigroup Theory is in [24] and [34].In Chapter 2 we show that for every monoid S which is given by a finite and complete presentation P = P[x, r], and for every n ~ 2, there is a chain of CW-complexes such that ~n has dimension n, for every 2 ~ s ~ n the s-skeleton of ~n is ~s and F acts on ~n. This action is called translation. Also we show that, for 2 ~ s ~ n, the open s-cells of ~n are in a 1-1 correspondence with the s-tuples of positive edges of V with the same initial. For the critical s-tuples, the corresponding open s-cells are denoted by Ps-I and the set of their open translates by F.Ps-I.F. The following holds true. if s ~ 3 if s = 2, where U stands for the disjoint union. Also, for every 2 ~ s ~ n - 1, there exists a cellular equivalence "'s on Ks = (~s X ~8)(s+1) such that Ks/ "'s= (V, PI, ... ,Ps-I) and the following is an exact sequence of (ZS, ZS)-bimodules where (D, Pl, ... , Ps-2) = V if s = 2. Using the above short exact sequences, we deduce that S is of type bi-FPn and that the free fi~ite resolution of'lS is S-graded. In Chapter 3 we generalize the notions left-(respectively right)-FPn and bi-FPn for small categories and show that bi-FPn implies left-(respectively right)-FPn . Also we show that another condition, which was introduced by Malbos and called FPn , implies bi-FPn . Since the name FPn is confusing, we call it here f-FPn for a reason which will be made clear in Section 3.1. Restricting to monoids, we show that, if a monoid is given by a finite and complete presentation, then it is of type f-FPn . Lastly, for every small category C, we show how to construct free resolutions of ZC, at lea..'lt up to dimension 3, using some geometrical ideas which can be generalized to construct free resolutions of ZC of any length. vi In Chapter 4 we study finiteness conditions of ~onoids of a combinatorial nature. We show that there are semigroups S in which min'R., is independent of other conditions which S may satisfy such as being finitely generated, periodic, inverse, E-unitary and even from the finiteness of the maximal subgroups of S. We also relate the congruences of a monoid with the finiteness condition minQ, and show that, if S is a monoid which satisfies minQ, then every congruence JC on S which contains Q is of finite index in S. If a semigroup satisfies minQ and has all its maximal subgroups locally finite, then we show that it is finite. Lastly, we show that, for trees of completely O-simple semigroups, the local finiteness of its maximal subgroups implies the local finiteness of the semigroups.

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.

This dissertation will seek to explore if the properties of some of the key results on semigroups and their compacti cations under the operation of addition also apply under the operation of multiplication. Consider- able emphasis will be placed on the semigroup N of the set of natural numbers and its compacti cation N. Furthermore, the dissertation will discuss the smallest ideal of a semi- group and highlight some of its fundamental properties. / Mathematics / M. Sc. (Mathematics)