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

Abstract

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 $(\overline M, \overline r)$ on the least cancellative image $\overline M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\overline r$ is an extension of $r$. It is also shown that non-degenerate irretractable solutions necessarily are bijective.