Using groups for investigating rewrite systems

TL;DR

This paper introduces a group-theoretic framework using monoids and groups to analyze and solve problems related to algebraic rewrite systems, enhancing methods for constructing algebras and addressing the word problem.

Contribution

It presents a novel approach that employs groups derived from rewrite system relations to investigate algebraic laws, offering practical tools for algebra construction and word problem solutions.

Findings

Efficient tools for analyzing rewrite systems are developed.

The approach facilitates algebra construction satisfying specific laws.

A practical method for solving the word problem is proposed.

Abstract

We describe several technical tools that prove to be efficient for investigating the rewrite systems associated with a family of algebraic laws, and might be useful for more general rewrite systems. These tools consist in introducing a monoid of partial operators, listing the monoid relations expressing the possible local confluence of the rewrite system, then introducing the group presented by these relations, and finally replacing the initial rewrite system with a internal process entirely sitting in the latter group. When the approach can be completed, one typically obtains a practical method for constructing algebras satisfying prescribed laws and for solving the associated word problem.