The Dushnik-Miller dimension of a partially-ordered set P is the smallest d such that one can embed P into a product of d linear orders. We prove that the dimension of the divisibility order on the interval $\{1,\dots ,n\}$, is equal to ${(\log n)}^2{(\log \log n)}^{-\mathrm{\Theta }(1)}$ as n goes to infinity.

We prove similar bounds for the 2-dimension of divisibility in $\{1,\dots ,n\}$, where the 2-dimension of a poset P is the smallest d such that P is isomorphic to a suborder of the subset lattice of $[d]$. We also prove an upper bound for the 2-dimension of posets of bounded degree and show that the 2-dimension of the divisibility poset on the set $(\alpha n,n]$ is ${\mathrm{\Theta }}_{\alpha }(\log n)$ for $\alpha \in (0,1)$. At the end we pose several problems.

部分有序集合P的Dushnik-Miller维是最小的d，因此可以将P嵌入到d个线性阶积中。我们证明了区间上的除数阶的维$\{\mathrm{1个}，\dots ，ñ\}$，等于 ${（\mathrm{日志}ñ）}^2{（\mathrm{日志}\mathrm{日志}ñ）}^{-\mathrm{\Theta }（\mathrm{1个}）}$当n达到无穷大时。

我们证明了2维除数的相似界 $\{\mathrm{1个}，\dots ，ñ\}$，其中位姿P的2维维是最小的d，使得P同构为P的子集晶格的子阶$[d]$。我们还证明了有界度的二维态的一个上限，并证明了该集上可除性二维集的二维$（\alpha ñ，ñ]$ 是 ${\mathrm{\Theta }}_{\alpha }（\mathrm{日志}ñ）$ 对于 $\alpha \in （0，\mathrm{1个}）$。最后，我们提出了几个问题。