Introduction

The purpose of this paper is first to give a survey of some recent results concerning semigroup presentations, and then to prove a new result which enables us to describe the structure of semigroups defined by certain presentations.

The main theme is to relate the semigroup S defined by a presentation II to the group G defined by II. After mentioning a result of Adjan's giving a sufficient condition for S to embed in G, we consider some cases where S maps surjectively (but not necessarily injectively) onto G. In these examples, we find that S has minimal left and right ideals, and it turns out that this is a sufficient condition for S to map onto G. In this case, the kernel of S (i.e. the unique minimal two-sided ideal of S) is a disjoint union of pairwise isomorphic groups, and we describe a necessary and sufficient condition for these groups to be isomorphic to G.

We then move on and expand on these results by proving a new result (Theorem 9), which is a sort of rewriting theorem, enabling us to determine the presentations of the groups in the kernel in certain cases. We finish off by applying this new result to certain semigroup presentations and by pointing out its limitations.