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...