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.

中文翻译：

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有限阶Yang-Baxter方程解的匹配乘积

在这项工作中，我们专注于Yang-Baxter方程的集理论解，该解是有限阶的，不一定是双射的。我们使用匹配的解决方案产品作为处理这些有限阶解决方案的统一工具，其中还包括对合和幂等解。尤其是，我们证明了两个解\（r_S \）和\（r_T \）的匹配乘积是且仅当\（r_S \）和\（r_T \）是有限阶的。此外，我们证明了在\（r_S \）和\（r_T \）上具有足够的信息时，我们可以精确地确定匹配乘积的顺序。最后，我们证明如果B是一个有限半括号，那么对于与B紧密相连的整数n，关联的解r满足\（r ^ n = r \）。