Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, $\mathcal{H}$ in SupLat, has a corresponding group, $R(\mathcal{H})$, which we call its remnant and a co-quasitriangular structure on $\mathcal{H}$ induces a YBE solution on $R(\mathcal{H})$, which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, $\mathcal{H}$, this secondary group structure appears as the projection of the transmutation of $\mathcal{H}$. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution.

斜括号最近作为一种研究杨-巴克斯特方程的集合论解的方法引起了人们的注意。在这里，我们通过研究完全格和连接保留态射类别 SupLat 中的 Hopf 代数，提出了这些解决方案的新方法。我们通过展示任何 Hopf 代数来连接这两种方法，$\mathcal{H}$ 在 SupLat 中，有一个对应的组， $\mathrm{电阻}(\mathcal{H})$，我们称其为残差和协拟三角结构 $\mathcal{H}$ 诱导 YBE 解决方案 $\mathrm{电阻}(\mathcal{H})$，这与其组结构兼容。相反，任何具有兼容 YBE 解决方案的组都可以通过这种方式实现。此外，众所周知，任何此类组都具有诱导的二级组结构，使其成为偏左括号。通过将群实现为协拟三角 Hopf 代数的余数，$\mathcal{H}$，这种二级基团结构表现为 $\mathcal{H}$. 最后，对于任何 YBE 解，我们在 SupLat 中获得了 SupLat-F​​RT Hopf 代数，其残差恢复了解的通用斜撑。