The Dimension of Divisibility Orders and Multiset Posets

Abstract

AbstractThe 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 $$[N/\kappa , N]$$ [ N / κ , N ] is bounded above by $$\kappa (\log \kappa )^{1+o(1)}$$ κ ( log κ ) 1 + o ( 1 ) and below by $$\Omega ((\log \kappa /\log \log \kappa )^2)$$ Ω ( ( log κ / log log κ ) 2 ) . We improve the upper bound to $$O((\log \kappa )^3/(\log \log \kappa )^2).$$ O ( ( log κ ) 3 / ( log log κ ) 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.