Amalgams of Inverse Semigroups and C*-algebras

An amalgam of inverse semigroups [S,T,U] is full if U contains all of the idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam of C*(S) and C*(T) over C*(U). Using this result, we describe certain amalgamated free products of C*-algebras, including finite-dimensional C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs

Derivations and relation modules for inverse semigroups

We define the derivation module for a homomorphism of inverse semigroups, generalizing a construction for groups due to Crowell. For a presentation map from a free inverse semigroup, we can then define its relation module as the kernel of a canonical map from the derivation module to the augmentation module. The constructions are analogues of the first steps in the Gruenberg resolution obtained from a group presentation. We give a new proof of the characterization of inverse monoids of cohomological dimension zero, and find a class of examples of inverse semigroups of cohomological dimension one

Graph automatic semigroups

In this thesis we examine properties and constructions of graph automatic semigroups, a generalisation of both automatic semigroups and finitely generated FA-presentable semigroups. We consider the properties of graph automatic semigroups, showing that they are independent of the choice of generating set, have decidable word problem, and that if we have a graph automatic structure for a semigroup then we can find one with uniqueness. Semigroup constructions and their effect on graph automaticity are considered. We show that finitely generated direct products, free products, finitely generated Rees matrix semigroup constructions, zero unions, and ordinal sums all preserve unary graph automaticity, and examine when the converse also holds. We also demonstrate situations where semidirect products, Bruck-Reilly extensions, and semilattice constructions preserve graph automaticity, and consider the conditions we may impose on such constructions in order to ensure that graph automaticity is preserved. Unary graph automatic semigroups, that is semigroups which have graph automatic structures over a single letter alphabet, are also examined. We consider the form of an automaton recognising multiplication by generators in such a semigroup, and use this to demonstrate various properties of unary graph automatic semigroups. We show that infinite periodic semigroups are not unary graph automatic, and show that we may always find a uniform set of normal forms for a unary graph automatic semigroup. We also determine some necessary conditions for a semigroup to be unary graph automatic, and use this to provide examples of semigroups which are not unary graph automatic. Finally we consider semigroup constructions for unary graph automatic semigroups. We show that the free product of two semigroups is unary graph automatic if and only if both semigroups are trivial; that direct products do not always preserve unary graph automaticity; and that Bruck-Reilly extensions are never unary graph automatic

Automatic S-acts and inverse semigroup presentations

To provide a general framework for the theory of automatic groups and semigroups, we introduce the notion of an automatic semigroup act. This notion gives rise to a variety of definitions for automaticity depending on the set chosen as a semigroup act. Namely, we obtain the notions of automaticity, Schutzenberger automaticity, R- and L-class automaticity, etc. We discuss the basic properties of automatic semigroup acts. We show that if S is a semigroup with local right identities, then automaticty of a semigroup act is independent of the choice of both the generators of S and the generators of the semigroup act. We also discuss the equality problem of automatic semigroup acts. To give a geometric approach, we associate a directed labelled graph to each S-act and introduce the notion of the fellow traveller property in the associated graph. We verify that if S is a regular semigroup with finitely many idempotents, then Schutzenberger automaticity is characterized by the fellow traveller property of the Schutzenberger graph. We also verify that a Schutzenberger automatic regular semigroup with finitely many idempotents is finitely presented. We end Chapter 3 by proving that an inverse free product of Schutzenberger automatic inverse semigroups is Schutzenberger automatic. In Chapter 4, we first introduce the notion of finite generation and finite presentability with respect to a semigroup action. With the help of these concepts we give a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated and finitely presented as an inverse semigroup. We end Chapter 4 by giving a necessary and sufficient condition for the semidirect product of a semilattice by a group to be Schutzenberger automatic. Chapter 5 is devoted to the study of HNN extensions of inverse semigroups from finite generation and finite presentability point of view. Namely, we give necessary and sufficient conditions for finite presentability of Gilbert's and Yamamura's HNN extension of inverse semigroups. The majority of the results contained in Chapter 5 are the result of a joint work with N.D. Gilbert and N. Ruskuc

Subgroups of free idempotent generated semigroups need not be free

We use topological methods to study the maximal subgroups of the free idempotent generated semigroup on a biordered set. We use these to give an example of a free idempotent generated semigroup with maximal subgroup isomorphic to the free abelian group of rank 2. This is the first example of a non-free subgroup of a free idempotent generated semigroup

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update