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We study diagonal mappings in non-involutive set-theoretic solutions of the Yang-Baxter equation. We show that they are commuting bijections. We also give equational characterization of multipermutation solutions and extend results of Rump and Gateva-Ivanova about decomposability to non-involutive solutions. We show that each, not necessarily involutive, square-free multipermutation solution of arbitrary cardinality, is always decomposable.
Jedlička et al. (Fri,) studied this question.
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