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arXiv · 2201.12952

The Dimension of Divisibility Orders and Multiset Posets

Milan Haiman
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Abstract

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 showed that the dimension of the divisibility 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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Milan Haiman. 2022-01-31. The Dimension of Divisibility Orders and Multiset Posets. https://arxiv.org/abs/2201.12952

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