The Yang-Baxter equation and Thompson's group $F$

TL;DR

This paper introduces partial set-theoretic solutions to the Yang-Baxter equation, explores their algebraic structures, and demonstrates a connection to Thompson's group F through a specific example.

Contribution

It extends the theory of Yang-Baxter solutions to partial cases and links these structures to Thompson's group F, providing new algebraic insights.

Findings

Partial solutions can be embedded into inverse monoids.

Existence of a partial solution with structure group isomorphic to Thompson's group F.

Square-free partial solutions have specific algebraic embeddings.

Abstract

In analogy with non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation and braces, we define non-degenerate involutive partial set-theoretic solutions and partial braces. We define the structure group and the structure inverse monoid of such a solution and prove that if the partial solution is square-free, then its structure inverse monoid embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there exists a square-free, non-degenerate involutive partial solution with structure group isomorphic to Thompson's group $F$.