# The matched product of the solutions to the Yang-Baxter equation of finite order

## TL;DR

This paper investigates finite order solutions to the set-theoretical Yang-Baxter equation using matched products, establishing conditions for finite order preservation and linking solutions to algebraic structures like semi-braces.

## Contribution

It introduces the matched product as a unifying framework for finite order solutions and characterizes their order in relation to component solutions and semi-braces.

## Key findings

- Matched product of solutions preserves finite order.
- Order of the matched product depends on component solutions.
- Solutions from finite semi-braces satisfy a specific idempotent relation.

## Abstract

In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions $r_S$ and $r_T$ is of finite order if and only if $r_S$ and $r_T$ are. Furthermore, we show that with sufficient information on $r_S$ and $r_T$ we can precisely establish the order of the matched product. Finally, we prove that if $B$ is a finite semi-brace, then the associated solution $r$ satisfies $r^n=r$, for an integer $n$ closely linked with $B$.