Primes in coverings of indecomposable involutive set-theoretic solutions to the Yang-Baxter equation

Abstract

Jan Okniński raised the question whether the primes dividing the size $n$ of a finite indecomposable set-theoretic solution to the Yang-Baxter equation are related to the primes dividing the order of the associated permutation group. With Cedó he proved that both prime sets are equal if $n$ is square-free. We characterize equality and prove that surjective morphisms of solutions admit a canonical factorization into a covering and a morphism given by a brace ideal. The existence of solutions with non-equality of the prime sets is reduced to irretractable solutions. It is proved that non-equality is possible, and a minimal example is constructed..