Algorithms for Garside calculus

TL;DR

This paper explores algorithms based on Garside calculus, extending its application to categories and Garside families, to effectively solve the Word Problem in these generalized algebraic structures.

Contribution

It introduces and addresses algorithmic questions within the extended Garside framework, broadening the scope of Garside calculus beyond traditional contexts.

Findings

Algorithms effectively solve the Word Problem in categories with Garside structures.

Extension of Garside calculus to new algebraic contexts demonstrated.

Enhanced understanding of normal forms in generalized monoids and groups.

Abstract

Garside calculus is the common mechanism that underlies a certain type of normal form for the elements of a monoid, a group, or a category. Originating from Garside's approach to Artin's braid groups, it has been extended to more and more general contexts, the latest one being that of categories and what are called Garside families. One of the benefits of this theory is to lead to algorithms solving effectively the naturally occurring problems, typically the Word Problem. The aim of this paper is to present and solve these algorithmic questions in the new extended framework.