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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 thedimension 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 resulton posets of multisets ordered by inclusion. We also consider otherdivisibility orders and give a bound for polynomials ordered by divisibility.

The Dimension of Divisibility Orders and Multiset Posets