Diagonals of solutions of the Yang–Baxter equation

Abstract We study the diagonal mappings in non-involutive set-theoretic solutions of the Yang–Baxter equation. We show that, for non-degenerate solutions, they are commuting bijections. This gives the positive answer to the question: “Is every non-degenerate solution bijective?” of Cedó, Jespers and Verwimp. Additionally, we show that, for a subclass of solutions called k - permutational , only one-sided non-degeneracy suffices to prove that one of the diagonal mappings is invertible. We also present an equational characterization of multipermutation solutions and extend results of Rump, Gateva–Ivanova and Castelli, Mazzotta, Stefanelli about decomposability to non-involutive infinite case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable.