Structure monoids of set-theoretic solutions of the Yang-Baxter equation

Given a set-theoretic solution (X, r) of the Yang–Baxter equation, we denote by M = M(X, structure monoid and A A(X, r), respectively A0 left, right, derived r). It is shown that there exist left action on right 1-cocycles π 0 with coefficients in respect to these actions, respectively. We investigate when are injective, surjective, or bijective. In case X finite, it turns out bijective if only non-degenerate, non-degenerate. particular bijective, define semi-truss then show this naturally induces (M, least cancellative image r)/η embedded r)/η, for example irretractable, r an extension r. also non-degenerate irretractable solutions necessarily