The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

In this work, we focus on the set-theoretical solutions of the Yang–Baxter equation which are of finite order and not necessarily bijective. 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 r _S and r _T is of finite order if and only if r _S and r _T are. Furthermore, we show that with sufficient information on r _S and r _T, we can precisely establish the order of the matched product. Finally, we prove that if B is a finite semi-brace, then the associated solution r satisfies r ⁿ= r, for an integer n closely linked with B.