Matched pairs approach to set theoretic solutions of the Yang–Baxter equation

We study 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. We give a characterisation of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid S(X,r) and we show that r extends as a solution rS on S(X,r) as a set. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids S of the required type, including iterated products S⋈S⋈S equivalent to rS a solution, and extensions (S⋈T,rS⋈T). Examples include a general ‘double’ construction (S⋈S,rS⋈S) and some concrete extensions, their actions and graphs based on small sets.