The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order
The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order
复制标题
有限阶Yang-Baxter方程解的匹配积
DOI：
10.1007/s00009-020-1483-y
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发表时间：
2019
影响因子：
1.1
通讯作者：
Paola Stefanelli
中科院分区：
文献类型：
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作者：
F. Catino;I. Colazzo;Paola Stefanelli
在这项工作中，我们关注Yang-Baxter方程的有限阶和不一定是双射的集合理论解。我们用解的匹配积作为统一的工具来处理这些有限阶的解，也包括对合解和幂等解。特别地，我们证明了两个解$$r_S$$ r S和$$r_T$$ r T的匹配积是有限阶的当且仅当$$r_S$$ r S和$$r_T$$ r T是。进一步，我们证明了有足够的$$r_S$$ r S和$$r_T$$ r T的信息，我们可以精确地建立匹配产品的顺序。最后，我们证明了如果B是有限半括号，那么对于与B紧密相连的整数n，其相关解r满足$$r^n=r$$ r n = r。
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 $$r_S$$ r S and $$r_T$$ r T is of finite order if and only if $$r_S$$ r S and $$r_T$$ r T are. Furthermore, we show that with sufficient information on $$r_S$$ r S and $$r_T$$ 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 $$r^n=r$$ r n = r , for an integer n closely linked with B .