The matched product of set-theoretical solutions of the Yang-Baxter equation

In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bi- jective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions rS and rT is of finite order if and only if rS and rT are. Furthermore, we show that with sufficient information on rS and rT wecanpreciselyestablishtheorderofthematchedproduct.Finally, we prove that if B is a finite semi-brace, then the associated solution r satisfies rn = r, for an integer n closely linked with B.

Publication year:2020

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