The Dimension of Divisibility Orders and Multiset Posets

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 isibility 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.