For the lattices of all subsets of a given set, an axiomatic characterization is known: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra.

The question: How to characterize the sets of filters on a set? That is, having a poset, how to check whether it is isomorphic to the set of filters on some set?

Note that we allow improper filters (An improper filter is also filter) to ensure that the set of filters is a complete lattice.