Length functions for semigroup embeddings

Abstract

Following the work done in [O] for groups, we describe, for a given semigroup S, which functions l:S→N can be realized up to equivalence as length functions g↦|g|H by embedding S into a finitely generated semigroup H. We also, following the work done in [O2] and [OS], provide a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup