可除性序与多重集偏序的维数
The Dimension of Divisibility Orders and Multiset Posets
AI总结：
本文研究偏序集的Dushnik-Miller维数，改进了整数区间上可除性序的维数上界至O((log κ)^3/(log log κ)^2)，并推广到多重集偏序及多项式可除性序。
摘要：
偏序集 $P$ 的 Dushnik--Miller 维数是使得 $P$ 能嵌入到 $d$ 条链的乘积中的最小 $d$。Lewis 和 Souza 证明了整数区间 $[N/κ, N]$ 上的可除性序的维数上界为 $κ(\logκ)^{1+o(1)}$，下界为 $Ω((\logκ/\log\logκ)^2)$。我们将上界改进为 $O((\log κ)^3/(\log\logκ)^2)$。我们从一个关于按包含序排列的多重集偏序集的更一般结果推导出这个界。我们还考虑了其他可除性序，并给出了按可除性排序的多项式的界。
英文摘要：
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 $[N/κ, N]$ is bounded above by $κ(\logκ)^{1+o(1)}$ and below by $Ω((\logκ/\log\logκ)^2)$. We improve the upper bound to $O((\log κ)^3/(\log\logκ)^2).$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
