Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations $L_x$ are bijections, the squaring map is a bijection, and the identity $(xy)(xz) = (yx)(yz)$ holds. We call these algebras \emph{rumples} in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order $n$ exists if and only if $n=p_1^{p_1 k_1}\cdots p_m^{p_m k_m}$ for some distinct primes $p_i$ and positive integers $k_i$. A large class of affine solutions is obtained from nonsingular near-circulant matrices $A$, $B$ satisfying $[A,B]=A^2$. We characterize affine latin rumples as those latin rumples for which the displacement group generated by $L_x L_y\inv$ is abelian and normal in the group generated by all translations. We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity $(zx)(yx) = (zy)(xy)$ holds as well, and we show, among other results, that the generators $L_x L_y\inv$ of their displacement group have order dividing four.

中文翻译：

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Yang-Baxter 方程的对合拉丁解

Wolfgang Rump 证明了 Yang-Baxter 方程的非退化对合集合论解与二元代数之间存在一一对应的关系，其中所有左平移 $L_x$ 都是双射，平方映射是双射，恒等式$(xy)(xz) = (yx)(yz)$ 成立。我们称这些代数为 \emph{rumples}，类似于 quandles，这是另一类给出 Yang-Baxter 方程解的二元代数。我们专注于拉丁文的 rumles，即所有正确翻译也是双射的 rumples。我们证明 $n$ 阶仿射拉丁文隆波存在当且仅当 $n=p_1^{p_1 k_1}\cdots p_m^{p_m k_m}$ 对于某些不同的素数 $p_i$ 和正整数 $k_i$。一大类仿射解是从满足 $[A,B]=A^2$ 的非奇异近循环矩阵 $A$、$B$ 中获得的。我们将仿射拉丁皱褶描述为那些由 $L_x L_y\inv$ 生成的位移群在所有平移生成的群中是阿贝尔和正态的拉丁皱褶。我们充分发展了皱褶的可拓理论，以获得与群不仿射，甚至不是同位素的拉丁皱褶的例子。最后，我们研究了拉丁文皱褶，其中双重身份 $(zx)(yx) = (zy)(xy)$ 也成立，并且我们在其他结果中展示了它们位移的生成器 $L_x L_y\inv$组有顺序除四。