Diagonals of solutions of the Yang-Baxter equation

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 Ced\'{o}, 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 and Gateva-Ivanova about decomposability to non-involutive case. In particular, we show that each, not necessarily involutive, square-free multipermutation solution of finite level and arbitrary cardinality, is always decomposable.