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 $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, 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 $A'$ and 1-cocycles $\pi$ and $\pi'$ of $M$ with coefficients in $A$ and in $A'$ 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 $\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r)$ is left non-degenerate, in particular $\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\bar M, \bar r)$ on the least cancellative image $\bar M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.