Given a distributive lattice $A$ we can look at $Spec(A)$, whose points are prime ideals and its open sets are given similarly to the Zariski topology on Spec of a ring. That is, the basis of open sets is composed of sets of the form $D(I)=\{p~\mathrm{prime~in~A}:I\nsubseteq p\}$.

So, given a prime ideal, it is not hard to show that its complement is a prime filter. Hence there is a set bijection between the set of prime ideals and the set of prime filters. Does anyone know, if we force this bijection to be a homeomorphism based on the topology on $Spec(A)$, is there a nice description of the open basis elements on the set of prime filters of $A$?

Note: Perhaps this question is purely lattice theory. I guess it depends on your point of view. Please add or remove tags as necessary.

Thanks!

Jon