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Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd-Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program. Definitions and preliminary theoremsLet X be a finite set and let F S (X) be the free semigroup (excluding the empty word) on X. Throughout this paper, by a presentation P on X we mean a generating set X together with a finite set of relations L ; = R h i e / where L h R t e F S (X). We write and use the notation SmgpP and GpP for the free-est semigroup on X satisfying the relations L, = i?,, iel and the free-est group on X satisfying the relations L, = /?,, iel, respectively. We are thus only interested in finitely presented semigroups and finitely presented groups. We use the symbols F*J and Z for the set of positive integers and the set of integers, respectively. Definition 1.1. Let P be a presentation on X. We say that SmgpP has a group kernel if it has an ideal which is isomorphic to Gp P. Theorem 1.2. Let P be a presentation, S = SmgpP and G = GpP. (a) S has a group kernel if and only if S has an idempotent e such that Se is a group which is an ideal in S.

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The index and period of an element a of a finite semigroup are the smallest values of m ^ 1 and r ^ 1 such that a m+T = a m . An element with index m and period 1 is called an m-potent element. For an element a of a finite full transformation semigroup with index m and period r, a unique factorisation a = of} such that Shift(cr) PI Shift(/?) = 0 is obtained, where a is a permutation of order r and 0 is an m-potent. Some applications of this factorisation are given.