Direct Sum Decompositions of Matroids and Exponential Structures
Direct Sum Decompositions of Matroids and Exponential Structures
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拟阵和指数结构的直和分解
DOI：
10.1006/jctb.1995.1017
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
V. Welker
中科院分区：
文献类型：
--
作者：
V. Welker
我们将其关联到一个简单的拟阵（或几何格）M和划分M的秩的数d是偏序集Dd（M），其上区间是（集-）划分格。事实上，在某些重要的情况下，它们是斯坦利意义上的指数结构[11]。我们的建设包括分区格，偏序集的分区的大小是由一个固定的数字d整除，和偏序集的直接和分解的有限向量空间。若M是模可补拟阵，则偏序集Dd（M）是CL-可壳的。这推广了Sagan和Wachs的结果，解决了直和分解偏序集的可壳性问题。我们分析炮击，并得出一些事实的下降链。我们可以应用这些技巧来检索Wachs关于d-可分划分格中降链的结果。我们还得到了一个公式的莫比乌斯数的偏序集的直和分解的向量空间。
We associate to a simple matroid (resp. a geometric lattice) M and a number d dividing the rank of M a partially ordered set Dd(M) whose upper intervals are (set-) partition lattices. Indeed, for some important cases they are exponential structures in the sense of Stanley 11]. Our construction includes the partition lattice, the poset of partitions whose size is divisible by a fixed number d, and the poset of direct sum decompositions of a finite vector space. If M is a modularly complemented matroid the posets Dd(M) are CL-shellable. This generalizes results of Sagan and Wachs and settles the open problem of the shellability of the poset of direct sum decompositions. We analyse the shelling and derive some facts about the descending chains. We can apply these techniques to retrieve the results of Wachs about descending chains in the lattice of d-divisible partitions. We also derive a formula for the Mobius number of the poset of direct sum decompositions of a vector space.