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Morita equivalence is a general way of classifying structures by means of their actions that is weaker than isomorphism but at the same time useful. It arose first in the study of unital rings in the 1950’s [35] but has since been extended to many other kinds of strucures, including classes of non- unital rings. It was first applied to semigroup theory in the 1970’s in the work of Banaschewski [5] and Knauer [19] who independently determined when two monoids were Morita equivalent. However they were unable to extend their definition to arbitrary semigroups since Banaschewski showed that Morita equivalence reduced to isomorphism. It was not until the 1990’s that Talwar [40, 41] was able to find a good definition of Morita equivalence for a class of semigroups that included all monoids but also all regular semigroups: the class of semigroups with local units. Such a semigroup is one in which each element has a left and a right idempotent identity. Talwar’s work was not developed further until the twenty-first century when a variety of mathematicians including Funk, Laan, Lawson, M´arki and Steinberg started to develop the Morita theory of semigroups in detail [9, 20, 25, 39]. Our thesis takes as its starting point Lawson’s reinterpretation of Talwar’s work. The thesis consists of three chapters. An essential ingredient in Morita theory is the notion of an equivalence of categories. For this reason, Chapter 1 of this thesis reviews all the categorical definitions needed. In Chapter 2, we describe in detail the work of Banaschewski and Knauer on the Morita theory of monoids. These two chapters contain no new work. We begin Chapter 3 by explaining why the obvious way of defining the Morita equivalence of two semigroups does not work. We then describe Lawson’s approach to Talwar’s work. This provides the foundation for our thesis. Our new contributions to the theory are contained in Sections 3.2, 3.3 and 3.4 and are based on Rees matrix semigroups. Talwar showed that the classical Rees matrix theorem for completely simple semigroups could be regarded as a Morita theorem: a semigroup is Morita equivalent to a group if and only if it is completely simple if and only if it is isomorphic to a Rees matrix semigroup over a group. This raises the question of determining what role Rees matrix semigroups play in the Morita theory of semigroups with local units. We investigate three different problems based on this idea: Section 3.2 In this section, we try to provean exact generalisation of the Rees theorem. We are interested in the case where S is Morita equivalent to T if and only if S is isomorphic to some kind of Rees matrix semigroup over T. Section 3.3 In this section, we prove that S is Morita equivalent to T if and only if S is a locally isomorphic image of a special kind of Rees matrix semigroup over T. This result was first proved by Laan and M´arki [20] but we give a new proof that generalizes the classical proof of the Rees theorem. Section 3.4 Finally, we solve the following problem: given an inverse semi-group S find all inverse semigroups T which are Morita equivalent to S. Our solution uses special kinds of Rees matrix semigroups over S. In this section, we also describe those semigroups which are Morita equivalent to semigroups with commuting idempotents. This builds on early work by Khan and Lawson [17, 18].

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Universal algebra has long been regarded as a fundamental tool in studying semantics of programming languages. Within this paradigm, one can formulate statements regarding the correctness of a program by looking at the interpretations of the code in any model for the language.;While this provides a description of finite computations, other models have to be introduced in order to provide a semantics for recursion and infinite computations in general. This leads to a study of rational and infinite terms. Such terms arise by a dual construction to that of the finite ones. namely, while the latter form an initial algebra, the former are a final coalgebra.;For this reason, it is natural to approach the study of infinite terms by dualising the categorical model of universal algebra. This leads to various different constructions, which are worth of investigation. In this thesis we approach two of them. In one case, we introduce the notions of cosignature, coequation and comodel, in the spirit of the theory of coalgebraic specification. In the second we focus on the properties of monads which can model infinitary computations. Such monads we call guarded, and include, amongst others, the monads of finite terms, infinite terms, rational terms and term graphs. As a byproduct of identifying this notion, we can solve algebraic systems of equation, which are an abstract counterpart to the notion of a recursive program scheme.;Many guarded monads we encounter are obtained by collecting, in an appropriate sense, a suitable family of coagebras. These examples are all instances of a general theorem we present, which tells under which conditions we can define a monad by a colimit operation, and when such comonads are guarded.;The level of abstraction allowed by the use of the categorical formalism allows us to instantiate some of the results in different categories, obtaining a monadic semantics for rational and infinite parallel term rewriting.

We generalize the category O of Bernstein, Gelfand and Gelfand to the so called fat category O, On and derive some of its properties. From a Lie theoretic point of view, contains a significant amount of indecomposable representations that do not belong to O (although it fails to add new simple ones) such as the fat Verma modules. These modules have simple top and socle and may be viewed as standard objects once a block decomposition of is obtained and each block is seen to be equivalent to a category of finite dimensional modules over a finite dimensional standardly stratified algebra. We describe the Ringel dual of these algebras (concluding that principal blocks are self dual) and we obtain the character formulae for their tilting modules. Furthermore, a double centralizer property is proved, relating each block with the corresponding fat algebra of coinvariants. As a byproduct we obtain a classification of all blocks of in terms of their representation type. In the process of determining the quiver and relations which characterize the basic algebras associated to each block of On we prove (for root systems of small rank) a formula establishing the dimension of the Ext1 spaces between simple modules. By borrowing from Soergel some results describing the behaviour of the combinatorial functor V, we are able to compute examples.

In this thesis, the word "ring" will always mean "commutative, (associative,) Noetherian ring with a non-zero multiplicative identity". An arbitrary module M over a ring A determines in a natural, but mildly complicated, fashion a certain complex:-</p> <table> <tr> <td colspan="4"> </td> <td>d^-1 <td> </td> <td>d⁰ <td colspan="5"> </td> <td>dⁿ <td> </td> </tr> <tr> <td><strong>C(M)</strong>:</td> <td>0</td> <td>→</td> <td>M</td> <td>→</td> <td>M⁰</td> <td>→</td> <td>M¹</td> <td>→</td> <td>andhellip;</td> <td>→</td> <td>Mⁿ</td> <td>→</td> <td>andhellip;</td> </tr> </table> <p>of A-modules and -homomorphisms. This thesis is concerned with the construction of <strong>C(M)</strong>, with some of its properties and uses, and with some natural extensions of this theory. I have called <strong>C(M)</strong> the Cousin Complex for M: this seems appropriate because it is, in fact, the commutative algebra analogue of the Cousin Complex of andsect;2 of Chapter IV of Hartshorne (<strong>12</strong>). One can, of course, regard A as a module over itself and construct <strong>C(A)</strong>. The power of the Cousin Complex stems mainly from simple characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex. A ring A is Cohen-Macaulay if and only if <strong>C(A)</strong> is exact, and A is Gorenstein if and only if <strong>C(A)</strong> provides an injective (resp. minimal infective) resolution for A. Indeed, it was while looking for some sort of natural minimal injective resolution for a Gorenstein ring that I stumbled upon the existence of <strong>C(A)</strong> for an arbitrary ring A, and it was only later (considerably later, in fact, for my ability to work with algebraic geometry in general and schemes in particular is limited) that I realised <strong>C(A)</strong> was, in fact, the analogue of the Cousin Complex of the Affine scheme of the ring A as defined in Hartshorne (<strong>12</strong>). While familiarity with the fundamental ideas of commutative algebra and homological algebra is assumed, a survey of those basic standard results and definitions which are important for the subsequent work is given in the first two Chapters: Chapter 1 is devoted to commutative algebra and Chapter 2 to homological algebra. The main purposes of this are to establish notation and to attempt to leave the way open for comparatively unimpeded progress through the discussion of the Cousin Complex and its applications in Chapters 3-12. The experienced reader may like to begin at Chapter 3 and refer to Chapters 1 and 2 when necessary. The construction and properties of the Cousin Complex provide the subject matter of Chapter 3, and Chapters 4 and 5 contain the characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex. I have, of course, made considerable use of the existing literature on Cohen-Macaulay rings and Gorenstein rings, and two papers to which I have frequently referred are Rees (<strong>24</strong>) and Bass(<strong>5</strong>). Chapter 4 actually examines the Cousin Complex for Cohen-Macaulay modules: the generalization from rings to modules is, in this case, fairly straightforward, and I have given the theory in the more general case. While several authors have studied Cohen-Macaulay modules (for example, see Chap. IV of Serre (<strong>26</strong>)), I have found no mention of "Gorenstein module" in the literature, and I have defined Gorenstein modules (over a ring A) as those non-zero, finitely-generated A-modules M for which <strong>C(M)</strong> provides a minimal injective resolution. In Chapter 6, I have examined some of the properties of Gorenstein modules and developed various characterizations of them: I have looked for characterizations which reduce to some of Bass' characterisations of Gorenstein rings in the particular case when m = A. (See Fundamental Theorem of andsect;1 of Bass (<strong>5</strong>).) This has not been as easy as eight have been expected, mainly for the following reason. Suppose for the moment that A is a local ring. In (3.3) of (<strong>5</strong>), Bass showed that, if M is a non-zero, finitely-generated A-module of finite injective dimension, then inj.dim._AM = codh_AA (not necessarily codh_AM). This result has particularly pleasant consequences when M = A: one is able to deduce that a local ring of finite injective dimension (as a module over itself) must be Cohen-Macaulay. However, in the situation of Bass' result mentioned above, it is not necessarily true that M is Cohen-Macaulay. Consequently, while Bass was able to characterize Gorenstein local rings as those local rings A for which inj.dim._AA is finite, there are non-zero, finitely-generated modules M over a local ring A which are not Gorenstein modules but for which inj.dim._AM is finite. The generalization of this characterization is more complicated and less obvious. Difficulties also arise when one attempts to generalize some of Bass' other characterizations of Gorenstein rings. Bass himself conjectured (also in (<strong>5</strong>)) that, for a local ring A, there exist non-zero, finitely-generated A-modules M of finite injective dimension only if A is Cohen-Macaulay. I have perhaps added some weight to this conjecture by showing that, if A is local and there exists a Gorenstein A-module, then A has to be a Cohen-Macaulay ring. The remarkably simple characterizations of Cohen-Macaulay rings and Gorenstein rings in terms of the Cousin Complex seem adequate reason for suspecting that the Cousin Complex may help in attempts to classify (commutative Noetherian) rings which are not necessarily Cohen-Macaulay. Chapter 7 contains a few brief skirmishes in this direction. As it seemed appropriate to examine the positions of Gorenstein and Cohen-Macaulay local rings in the general hierarchy of local rings, I have included a brief survey of some well-known special classes of local rings (Regular, Complete Intersections, Gorenstein, and Cohen-Macaulay) together with examples to illustrate the distinctions between these classes. Consequently, only a small part of the work in Chapter 7 is original. Chapter 8 is an illustration of the use of the Cousin Complex. The situation studied in (<strong>15</strong>) by Iversen is ideally suited for Cousin Complex arguments, and it turns out that the theory of the Cousin Complex yields more information than Iversen was able to obtain. As mentioned above, the construction of the Cousin Complex is mildly complicated. In certain situations it becomes desirable to compare two Cousin Complexes, or variations thereof, and these comparisons become very complicated. I have tried to describe these comparisons as simply and clearly as possible, and I feel that the results justify these somewhat unpleasant computations. One such comparison occurs in Chapter 9. Suppose A is a ring. In Chapter 6 it gradually becomes evident that it would be helpful to compare, for a Gorenstein A-module M and an x andisin; A which is such that xandsdot;M andne; M and x is not a zero-divisor on M, minimal injective resolutions for the A-module M and the A/(x)-module M/xM, and that the existing machinery is inadequate for this purpose. This problem is overcome in Chapter 9 by comparing the Cousin Complexes involved in a more general situation, and then using the fact the the Cousin Complex provides a minimal injective resolution for a Gorenstein module.

This thesis consists of two parts. Part I of this thesis is concerned with the Higman-Thompson group G2,1. We review and apply Definitions, Lemmas and Theorems described in a series of lectures delivered by Graham Higman during a visit to the Australian National University from July 1973 to October 1973 on a family of finitely presented infinite groups Gn,r for n 2 and r 1. This thesis will concentrate on the group G2,1 (otherwise know as Thompson’s group V). We give a brief account of the history of the Higman-Thompson group G2,1, we clarify the proof of the conjugacy problem for elements in quasi-normal form and we prove that the power conjugacy problem for the group G2,1 is decidable. Part II of this thesis concentrates on the existence and structure of mixed and unmixed Beauville p-groups, for p a prime. We start by exhibiting the first explicit family of mixed Beauville 2-groups and find the corresponding surfaces. We follow this up by exploring the method that was used to construct the family; this leads to further ramification structures for finite p-groups giving rise to surfaces isogenous to a higher product of curves. We finish by classifying the non-abelian Beauville pgroups of order p3, p4 and provide partial results for p-groups of order p5 and p6. We also construct the smallest Beauville p-groups for each prime p.

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