Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation

TL;DR

This paper explores the structure of finite left braces, using iterated matched products to construct new simple braces, thereby advancing the understanding of solutions to the Yang-Baxter equation.

Contribution

It introduces a method to decompose finite left braces via matched products, leading to the construction of new finite simple braces.

Findings

Constructed new families of finite simple left braces.

Provided a framework for decomposing finite braces into simpler components.

Enhanced the classification of solutions to the Yang-Baxter equation.

Abstract

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorphic, as a left brace, to $B$. It is hence of fundamental importance to describe all simple objects in the class of finite left braces. In this paper we focus on the matched product decompositions of an arbitrary finite left brace. This is used to construct new families of finite simple left braces.