标题： 二面体解的集合理论杨-巴克斯特方程 显示英文标题

标题： Dihedral solutions of the set theoretical Yang-Baxter equation

摘要： 我们引入了\emph{编织二面集}（BDS）的概念，用于描述Yang-Baxter方程（YBE）的集合论解，这些解在基础集的笛卡尔平方上提供了无限二面体群的表示。 导致在三个对象的对称群上产生表示的BDS称为\emph{编织三重集}（BTS）。 BDS的基本例子来自于对称空间。 我们证明拉丁BDS（LBDS）可以完全用唯一2可除Bruck环的对合来描述。 我们证明LBDS的同构类与唯一2可除Bruck环的对合共轭类一一对应。 我们描述了素数阶、素数平方阶和3倍素数阶的LBDS，直到同构。 使用\texttt{GAP}，我们枚举了27阶和81阶LBDS的同构类。 拉丁BTS，或LBTS，被证明与指数为3的交换Moufang环的对合一一对应（CML3），并且与LBDS类似，LBTS的同构类与CML3-对合的共轭类重合。 我们对最多81阶的LBTS进行了分类。 显示英文摘要

摘要： We introduce the notion of a \emph{braided dihedral set} (BDS) to describe set-theoretical solutions of the Yang-Baxter equation (YBE) that furnish representations of the infinite dihedral group on the Cartesian square of the underlying set. BDS which lead to representations of the symmetric group on three objects are called \emph{braided triality sets} (BTS). Basic examples of BDS come from symmetric spaces. We show that Latin BDS (LBDS) can be described entirely in terms of involutions of uniquely 2-divisible Bruck loops. We show that isomorphism classes of LBDS are in one-to-one correspondence with conjugacy classes of involutions of uniquely 2-divisible Bruck loops. We describe all LBDS of prime, prime-square and 3 times prime-order, up to isomorphism. Using \texttt{GAP}, we enumerate isomorphism classes of LBDS of orders 27 and 81. Latin BTS, or LBTS, are shown to be in one-to-one correspondence with involutions of commutative Moufang loops of exponent 3 (CML3), and, as with LBDS, isomorphisms classes of LBTS coincide with conjugacy classes of CML3-involutions. We classify all LBTS of order at most 81.