Shellable and Cohen-Macaulay partially ordered sets
Shellable and Cohen-Macaulay partially ordered sets
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DOI：
10.1090/s0002-9947-1980-0570784-2
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发表时间：
1980
影响因子：
1.3
通讯作者：
A. Björner
中科院分区：
文献类型：
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作者：
A. Björner
。在本文中，我们研究可壳偏序集（偏序集），即链的单纯复形是可壳的有限偏序集。结果表明，所有可接受的格子（包括所有有限半模和超可解格子）和所有有界局部半模有限偏序集都是可壳的。 R. Stanley 提出的一种用于标记某些格的哈斯图边缘的技术被推广到偏序集，并被证明暗示了可剥壳性，而斯坦利关于此类标记的 Jordan-Holder 序列的主要定理仍然有效。此外，我们展示了可以从其他可脱壳偏序集和复合体构造可脱壳偏序集的多种方法。这些结果产生了科恩-麦考利偏序集的几个新例子。例如，当且仅当 G 是超可解时，有限群 G 的子群格才是 Cohen-Macaulay（事实上可壳）。最后证明了有限平面分布格的所有高阶复形都是可壳化的。
. In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the Jordan-Holder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.