One-generator skew braces and indecomposable set-theoretic solutions to the Yang-Baxter equation

TL;DR

This paper investigates one-generator solutions to the Yang-Baxter equation, establishing their connection with finite skew braces, and provides new insights into involutive solutions with numerical examples for small cases.

Contribution

It extends the understanding of one-generator solutions, clarifies their relationship with skew braces, and addresses a question about indecomposable solutions.

Findings

Relationship between indecomposable solutions and finite skew braces established

Extended results to involutive and multipermutation solutions

Numerical results for small-sized solutions presented

Abstract

We study the class of one-generator solutions to the Yang-Baxter equation, extending some recent results concerning the classes of involutive and multipermutation solutions. Moreover we show the precise relationship between indecomposable solutions to the Yang-Baxter equation and finite one-generator skew braces, giving a positive answer to a question posed by Agata and Alicja Smoktunowicz. In the last part, we apply our results to the involutive case, and we present some numerical results involving solutions of small size.