Set-theoretic solutions to the Yang-Baxter equation of multipermutation level two are classified via transvection orbits which are abelian torsors. The relationship to square-free solutions and rack solutions is determined by using the square map and its relationship to non-degeneracy of the corresponding cycle set. The crucial role of left and right ideal powers of braces is discussed in connection with applications to higher multipermutation level. As an illustration, we give a simple proof of a recent theorem of Smoktunowicz on braces with nilpotent adjoint group, and a classification of braces with multipermutation level two, removing the finiteness in Smoktunowicz’ theorem for such braces.

多排列二阶杨-巴克斯特方程的集合论解是通过横对流轨道分类的，横对流轨道是阿贝尔托量。与无方解和机架解的关系是通过使用方图及其与相应循环集的非简并关系来确定的。大括号的左右理想幂的关键作用将结合更高的多排列级别的应用进行讨论。作为说明，我们给出了最近关于具有幂零伴随群的花括号的 Smoktunowicz 定理的简单证明，以及具有多重置换级别 2 的花括号的分类，消除了 Smoktunowicz 定理中对于此类花括号的有限性。