发表机构
University of Michigan, Ann Arbor, MI, USA（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进序多面体与链多面体的边定理，通过连通凸子集划分边并证明对应类基数相等，给出边数闭式公式，并揭示已知双射的结构，同时比较边方向、边长与直径，讨论对组合优化的影响。
AI 中文摘要
对于有限偏序集，我们将序多面体和链多面体的边划分为由偏序集的连通凸子集索引的类，并证明相应的类具有相同的基数。这给出了边数的公共闭式公式，并确定了Hibi、Li、Sahara和Shikama通过显式公式给出的双射是相应类之间简单双射的不相交并集，这解释了它为何是双射以及它如何作用于边的方向和长度。作为推论，这两个多面体具有相同数量的边方向，且重数匹配；链多面体平行于每个坐标子空间的边数至少与序多面体一样多；链多面体的边长被序多面体的边长所支配。当且仅当偏序集包含一个三元链时，后两个比较中出现严格不等式。我们讨论了这对理想和反链上的线性和凸组合优化的影响。我们还用可比图表示边数，其形式可推广到任意图的稳定集多面体，给出了级联偏序集的递推式以及分层、锯齿形和王冠偏序集的闭式公式，刻画了这两个多面体幺模等价的分发格，并证明了尽管这两个多面体具有相同数量的边，但其中任一个的直径可以任意大于另一个；另一方面，两个直径都以偏序集的宽度为界，并且对于序数和、级联偏序集以及锯齿形和王冠偏序集，它们重合。
英文摘要
For a finite poset, we partition the edges of the order polytope and of the chain polytope into classes indexed by the connected convex subsets of the poset, and we establish that corresponding classes have the same cardinality. This yields a closed formula for the common number of edges, and it identifies the bijection of Hibi, Li, Sahara and Shikama, given by them through an explicit formula, as a disjoint union of simple bijections between corresponding classes, which explains why it is a bijection and how it acts on edge directions and lengths. As consequences, the two polytopes have equally many edge directions, with matching multiplicities; the chain polytope has at least as many edges parallel to each coordinate subspace as the order polytope; and the edge lengths of the chain polytope are dominated by those of the order polytope. Strict inequality occurs in the last two comparisons exactly when the poset contains a three-element chain. We discuss implications for linear and convex combinatorial optimization over ideals and antichains. We also express the number of edges in terms of the comparability graph, in a form that extends to stable-set polytopes of arbitrary graphs, give a recursion for series-parallel posets and closed formulas for layered, zigzag and crown posets, characterize the distributive lattices for which the two polytopes are unimodularly equivalent, and demonstrate that, although the two polytopes have the same number of edges, either one can have the larger diameter, by an arbitrary amount; on the other hand, both diameters are bounded by the width of the poset, and they coincide for ordinal sums, series-parallel posets, and zigzag and crown posets.
