可除阶与多集序的维数

Order Pub Date : 2023-11-22 DOI:10.1007/s11083-023-09653-7
Milan Haiman
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引用次数: 2

摘要

偏序集P的Dushnik-Miller维数是P可以嵌入到d链的积中的最小d。整数区间\([N/\kappa , N]\)上的Lewis和Souza可见性阶上以\(\kappa (\log \kappa )^{1+o(1)}\)为界,下以\(\Omega ((\log \kappa /\log \log \kappa )^2)\)为界。我们将上界改进为\(O((\log \kappa )^3/(\log \log \kappa )^2).\)。我们从包含排序的多集的偏序集的一个更一般的结果中推导出这个上界。我们还考虑了其他可除性阶,并给出了可除性多项式的一个界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Dimension of Divisibility Orders and Multiset Posets

The Dushnik–Miller dimension of a poset P is the least d for which P can be embedded into a product of d chains. Lewis and Souza isibility order on the interval of integers \([N/\kappa , N]\) is bounded above by \(\kappa (\log \kappa )^{1+o(1)}\) and below by \(\Omega ((\log \kappa /\log \log \kappa )^2)\). We improve the upper bound to \(O((\log \kappa )^3/(\log \log \kappa )^2).\) We deduce this bound from a more general result on posets of multisets ordered by inclusion. We also consider other divisibility orders and give a bound for polynomials ordered by divisibility.

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