A note on set-theoretic solutions of the Yang-Baxter equation
A note on set-theoretic solutions of the Yang-Baxter equation
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关于 Yang-Baxter 方程的集合论解的注释
DOI：
10.1016/j.jalgebra.2016.04.015
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
A. Smoktunowicz
中科院分区：
文献类型：
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作者：
A. Smoktunowicz
本文证明了置换群G（X，r）的基数为立方自由数的Yang-Baxter方程的每一个有限非退化对合集理论解（X，r）都是多置换解.本文还研究了有限支撑的一些性质。还证明了如果A是基数为奇数的左支撑且对所有a，B∈ A有（− a）B=−（a B），则A是双边支撑，因而是Jacobson根环。还观察到有限多置换水平的花括号的半直积和圈积是有限多置换水平的花括号。
This paper shows that every finite non-degenerate involutive set theoretic solution (X, r) of the Yang–Baxter equation whose permutation group G (X, r) has cardinality which is a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated. It is also shown that if A is a left brace whose cardinality is an odd number and (− a)⋅ b=−(a⋅ b) for all a, b∈ A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level.