The first author was partially supported by the grants MINECO-FEDER MTM2017-83487-P and AGAUR 2017SGR1725 (Spain). The second author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium). The third author is supported by Fonds voor Wetenschappelijk Onderzoek (Flanders), via an FWO Aspirant-mandate.

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