Abstract

The use of graph theory has become widespread in the algebraic theory of semigroups. In this context, the graph is mainly used as a visual aid to make presentation clearer and the problems more manageable. Central to such approaches is the Cayley graph of a semigroup. There are also many variations on the idea of the Cayley graph, usually special kinds of subgraph or factor graph, that have become important in their own right. Examples include Schutzenberger graphs, Schreier coset graphs and Van Kampen diagrams (for groups), Munn trees, Adian graphs, Squier complexes, semigroup diagrams, and Graham-Houghton graphs of completely

*0*-simple semigroups. Also, the representation of elements in finite transformation semigroups as digraphs has proved a useful tool. This thesis consists of several problems in the theory of semigroups with the common feature that they are all best attacked using graph theory. The thesis has two parts. In the first part combinatorial questions for finite semigroups and monoids are considered. In particular, we look at the problem of finding minimal generating sets for various endomorphism monoids and their ideals. This is achieved by detailed analysis of the generating sets of completely*0*-simple semigroups. This investigation is carried out using the Graham-Houghton bipartite graph representation. The second part of the thesis is about infinite semigroup theory, and in particular some problems in the theory of semigroup presentations. We consider the general problem of finding presentations for subsemigroups of finitely presented semigroups. Sufficient conditions are introduced that force such a subsemigroup to be finitely presented. These conditions are given in terms of the position of the subsemigroup in the parent semigroup's left and right Cayley graphs.