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.