# A characterization of finite multipermutation solutions of the Yang-Baxter equation

## TL;DR

This paper characterizes finite multipermutation solutions of the Yang-Baxter equation by linking their structure groups to left orderability and poly-infinite cyclic properties, providing a clear algebraic criterion.

## Contribution

It establishes an equivalence between multipermutation solutions and the left orderability or poly-infinite cyclic nature of their structure groups.

## Key findings

- Finite multipermutation solutions correspond to structure groups that are left orderable.
- Such structure groups are exactly those that are poly-infinite cyclic.
- Provides an algebraic characterization of multipermutation solutions.

## Abstract

We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).