Geodesic rewriting systems and pregroups

TL;DR

This paper introduces geodesically perfect rewriting systems for groups and monoids, demonstrating their properties, construction methods, and applications using pregroups and Knuth-Bendix completion.

Contribution

It defines geodesically perfect systems, explores their properties, and connects them with pregroups and rewriting system construction methods.

Findings

Geodesically perfect systems are well-behaved and useful.

Construction methods include Knuth-Bendix completion and pregroups.

Examples provided for natural group presentations.

Abstract

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well-behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We describe a Knuth-Bendix completion process to construct such systems, show how they may be found with the help of Stallings' pregroups and conversely may be used to construct such pregroups.