REWRITING A SEMIGROUP PRESENTATION

Abstract

Let [Formula: see text] be a finitely presented semigroup having a minimal left ideal L and a minimal right ideal R. The main result gives a presentation for the group R∩L. It is obtained by rewriting the relations of [Formula: see text], using the actions of [Formula: see text] on its minimal left and minimal right ideals. This allows the structure of the minimal two-sided ideal of [Formula: see text] to be described explicitly in terms of a Rees matrix semigroup. These results are applied to the Fibonacci semigroups, proving the conjecture that S(r, n, k) is infinite if g.c.d.(n, k)>1 and g.c.d.(n, r+k−1)>1. Two enumeration procedures, related to rewriting the presentation of [Formula: see text] into a presentation for R∩L, are described. The first enumerates the minimal left and minimal right ideals of [Formula: see text], and gives the actions of [Formula: see text] on these ideals. The second enumerates the idempotents of the minimal two-sided ideal of [Formula: see text].