Ordering polytopes have been instrumental to the study of combinatorial optimization problems arising in a variety of fields including comparative probability, computational social choice, and group decision-making. The weak order polytope is defined as the convex hull of the characteristic vectors of all binary orders on n alternatives that are reflexive, transitive, and total. By and large, facet defining inequalities (FDIs) of this polytope have been obtained through simple enumeration and through connections with other combinatorial polytopes. This paper derives five new large classes of FDIs by utilizing the equivalent representations of a weak order as a ranking of n alternatives that allows ties; this connection simplifies the construction of valid inequalities, and it enables groupings of characteristic vectors into useful structures. We demonstrate that a number of FDIs previously obtained through enumeration are actually special cases of the large classes. This work also introduces novel construction procedures for generating affinely independent members of the identified ranking structures. Additionally, it states two conjectures on how to derive many more large classes of FDIs using the featured techniques.

排序多胞体对于研究各个领域中出现的组合优化问题发挥了重要作用，包括比较概率、计算社会选择和群体决策。弱阶多胞体被定义为n个自反、传递、全备选项上所有二元阶特征向量的凸包。总的来说，这个多胞体的刻面定义不等式（FDI）是通过简单的枚举和与其他组合多胞体的联系获得的。本文通过利用弱序的等效表示作为n的排名，推导了 5 个新的 FDI 大类允许联系的替代方案；这种联系简化了有效不等式的构造，并且能够将特征向量分组为有用的结构。我们证明，之前通过枚举获得的许多 FDI 实际上是大类的特例。这项工作还引入了新颖的构造程序，用于生成已识别的排名结构的密切独立的成员。此外，它还提出了关于如何使用特色技术派生更多大类 FDI 的两个猜想。