A combinatorial approach to noninvolutive set-theoretic solutions of the Yang-Baxter equation

We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself and in terms of the combinatorial properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$ and the graded quadratic $\textbf{k}$-algebra $A= A(\textbf{k}, X, r)$ over a field $\textbf{k}$. We investigate the class of (noninvolutive) square-free solutions $(X,r)$. It contains the particular class of self distributive solutions (i.e. quandles). We show that, similarly to the involutive case, every square-free braided set (possibly infinite, and not involutive) satisfies the cyclicity condions. We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We study an interesting class of finite square-free braided sets $(X,r)$ of order $n\geq 3$ which satisfy \emph{the minimality condition} M, that is $\dim_{\textbf{k}} A_2 =2n-1$ (equivalently, $A$ can be defined via exactly $(n-1)^2$ reduced binomial relations with special properties). In particular, every such solution is indecomposable. Examples are dihedral racks of prime order $p$. Finally, for general nondegenerate braided sets $(X,r)$ we discuss extensions of solutions with a special emphasis on \emph{strong twisted unions on braided sets}. We prove that if $(Z,r)$ is a nondegenerate 2-cancellative braided set which split as a strong twisted union $Z = X\natural Y$ of its $r$-invariant subsets $X$ and $Y$ then its braided monoid $S_Z$ is a strong twisted union $S_Z= S_X\natural S_Y$ of the braided monoids $S_X$ and $S_Y$. Moreover, if $(Z,r)$ is injective then its braided group $G_Z=G(Z,r)$ is also a strong twisted union $G_Z= G_X\natural G_Y$ of the associated braided groups of $X$ and $Y$.