Matched pairs approach to set-theoretic solutions of the Yang-Baxter equation

TL;DR

This paper investigates set-theoretic solutions to the Yang-Baxter equation using matched pair theory, characterizing involutive solutions and exploring extensions and constructions of solutions through monoids.

Contribution

It introduces a new framework linking matched pairs of monoids to solutions of the Yang-Baxter equation, including involutive and square-free cases, and develops methods for constructing and extending solutions.

Findings

Characterization of involutive square-free solutions via cyclicity conditions

Extension of solutions to associated monoids with matched pair properties

Construction of new solutions through iterated and double monoid products

Abstract

We study set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that $r$ extends as a solution $(S(X,r),r_S)$. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids $S$ of the required type, including iterated products $S\bowtie S\bowtie S$ equivalent to $r_S$ a solution, and extensions $(S\bowtie T,r_{S\bowtie T})$. Examples include a general `double' construction $(S\bowtie S,r_{S\bowtie S})$ and some concrete extensions, their actions and graphs based on small sets.