Length functions for semigroup embeddings

Abstract

Following the work done in [O] for groups, we describe, for a given semigroup \(S\), which functions \(l : S \to \mathbb{N} \) can be realized up to equivalence as length functions \(g\mapsto |g|_H\) 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.