  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.
