论文标题
划分命令和多板posets的维度
The Dimension of Divisibility Orders and Multiset Posets
论文作者
论文摘要
dushnik- poset $ p $的米尔尺寸是$ p $可以嵌入$ d $链的产品的最低$ d $。刘易斯（Lewis）和苏扎（Souza）表明，在整数间隔$ [n/κ，n] $的间隔上的划分顺序的维度在上面由$κ（\logκ）^{1+o（1）} $限制在$κ（\logκ/\ log -log -log -log -log -logκ）^2）$上。我们将上限提高到$ o（（（\logκ）^3/（\ log \logκ）^2）。$我们从更一般的结果中推断出这种结合，这是对包含序列的多动力的POSET。我们还考虑了其他可驱动性顺序，并为通过划分序列的多项式提供了界限。
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.
