Length Functions for Semigroup Embeddings

Abstract

Following the work done by Olshanskii 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 by Olshanskii and Sapir, provide a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup.