Rewriting Systems and Embedding of monoids in groups

TL;DR

This paper establishes conditions under which monoids with positive rewriting systems embed into groups, providing a simplified proof for right angled Artin monoids embedding into Artin groups.

Contribution

It introduces a new criterion linking rewriting systems to monoid embedding in groups, simplifying proofs for specific classes like right angled Artin monoids.

Findings

Monoids with certain positive rewriting systems embed in their groups.

A simplified proof for embedding right angled Artin monoids.

Reveals a connection between rewriting rules and algebraic embeddings.

Abstract

In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system $\Re$ that satisfies the condition that each rule in $\Re$ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from $\Re$ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris \cite{paris} that Artin monoids embed in their groups.