AI 中文总结
该研究用概率方法分析杨-巴克斯特方程的有限非退化集合论解与斜括号，引入四种衡量解偏离翻转程度的概率，揭示其刚性行为，还引入与不可分解分量、解接近程度相关的概率。
AI 中文摘要
我们采用概率方法研究杨-巴克斯特方程的有限非退化集合论解与斜括号。我们引入四种概率，分别从不同维度衡量解与翻转解的偏离程度。主要结果表明：对于由斜括号导出的解，这些概率呈现刚性行为——除了通过显式例子证明存在的有限个例外值外，它们均有略高于1/2的上界。在斜括号框架下，这些概率从两个不同维度衡量基础斜括号与平凡斜括号的偏离程度：一个通过零化子，另一个通过根基，还与几乎平凡斜括号相关。我们还引入两种概率：一种与解的不可分解分量相关，另一种衡量任意双射非退化映射成为解的接近程度。与主要结果不同，这两种概率在接近1时不呈现离散行为，可任意接近1。
英文摘要
We investigate finite non-degenerate set-theoretic solutions to the Yang--Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour --- apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ --- they can be made arbitrarily close to $1$.
Comments28 pages
