Braces, radical rings, and the quantum Yang–Baxter equation

Non-degenerate cycle sets are equivalent to non-degenerate unitary set-theoretical solutions of the quantum Yang–Baxter equation. We embed such cycle sets into generalized radical rings (braces) and study their interaction in this context. We establish a Galois theory between ideals of braces and quotient cycle sets. Our main result determines the relationship between two square-free cycle sets operating transitively on each other.