The Dimension of Divisibility Orders and Multiset Posets

Abstract

Abstract 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

$$[N/\kappa , N]$$

[ 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.