Set-theoretic solutions of the Yang-Baxter equation, graphs and computations
Let \(X\) be a nonempty set and let \(r: X \times X \to X \times X\) be a bijective map. The pair \((X,r)\) is called a quadratic set. The map \(r\) is square-free if \(r(x,y)= (x,y)\) for all \(x,y \in X\). The image of \((x,y)\) under \(r\) is presented as \(r(x,y)=(^xy,x^y)\). We write \(\mathcal L_x(y)= {}^xy, \mathcal R_x(y)=x^y\) for all \(x,y \in X\). The authors suppose that \(r\) is nondegenerate and square-free, that is \(r(x,x)=(x,x)\). A map \(r\) is a set-theoretic solution of the Yang-Baxter equation if the braid relation \(r^{12}r^{23}r^{12}=r^{23}r^{12}r^{23}\) holds in \(X \times X \times X\). To each quadratic set \((X,r)\) are associated canonical algebraic objects generated by \(X\) and with quadratic relations \(\mathfrak N=\mathfrak N(r)\) defined by \(xy=zt \in \mathfrak N(r)\), whenever \(r(x,y)=(z,t)\). The semigroup \(S=\langle X;\mathfrak N(r)\rangle\) (or group, or algebra) with a set of generators \(X\) and a set of defining relations \(\mathfrak N(r)\) is called a semigroup associated with \((X,r)\). The authors give a short historical review of the results on objects of such type (including the authors' papers), on the set-theoretic solutions of the Yang-Baxter equation and related structures. The goal of the paper is to extent the authors recent work on the set-theoretic solution of braid relation with new results about automorphism groups, strong twisted unions of solutions. The authors think that these results will allow them to introduce computational methods and develop algorithms for the case of multi-permutation square-free solutions of the Yang-Baxter equation. They can also be used in the construction of generalized twisted unions of solutions and in the classification of finite solutions of low multi-permutation level. The authors introduce graphical methods for the computation of solutions of finite order and their automorphisms. A detailed study of solutions multi-permutation level 2 is given.
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