Dehornoy's class and Sylows for set-theoretical solutions of the Yang-Baxter equation

TL;DR

This paper investigates the structure of cycle sets related to the Yang-Baxter equation, focusing on their decomposition into Sylow subgroups, Dehornoy's class, and Garside structures, providing new insights and methods.

Contribution

It introduces a decomposition approach for the structure group of cycle sets, explores Dehornoy's class, and offers a new method to retrieve Garside structures without relying on Rump's theorem.

Findings

Decomposition of structure groups into Sylow subgroups.

Bound on Dehornoy's class proved in specific cases.

Method to retrieve Garside structure without Rump's theorem.

Abstract

We explain how the germ of the structure group of a cycle set decomposes as a product of its Sylow-subgroups, and how this process can be reversed to construct cycle sets from ones with coprime classes. We study Dehornoy's class associated to a cycle set, and conjecture a bound that we prove in a specific case. We combine the use of braces and a monomial representation, in particular to answer a question by Dehornoy on retrieving the Garside structure without a theorem of Rump, while also retrieving said theorem.