Length Functions for Semigroup Embeddings

TL;DR

This paper characterizes which length functions can be realized through embeddings of semigroups into finitely generated or finitely presented semigroups, extending Olshanskii's work from groups to semigroups.

Contribution

It provides a complete description of length functions for semigroup embeddings and finitely generated semigroups, generalizing existing group results.

Findings

Characterization of length functions realizable via semigroup embeddings

Complete description of length functions in finitely generated semigroups

Extension of Olshanskii's group results to semigroups

Abstract

Following the work done by Olshanskii for groups, we describe, for a given semigroup $S$, which functions $l : S \rightarrow \mathbb{N}$ 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.