On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

Abstract This paper explores the structure groups G (X, r) of finite non-degenerate set-theoretic solutions (X, r) to the Yang–Baxter equation. Namely, we construct a finite quotient G (ₗ, ₑ) of G (X, r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G (X, r), then it also injects into G (ₗ, ₑ). We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G (X, r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.