# A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: 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.

• "Complete atomistic brouwerian lattice" is not enough: Consider any infinite atomistic boolean lattice, it is not a lattice of all filters on a set – porton Aug 16 '13 at 20:03

Under Stone-duality, the lattices which are isomorphic to the lattices of filters on a Boolean algebra are precisely the compact zero-dimensional frames. The lattices of filters on a complete Boolean algebra correspond to the compact zero-dimensional extremally disconnected frames. Therefore, by this answer, the lattices of filters on some set $X$ are precisely the compact zero-dimensional extremally disconnected frames where the isolated points in your frame form a dense subspace of the dual space in your frame. Translating this idea to a purely point-free context, the posets isomorphic to lattices of filters on a set are precisely the atomic compact zero-dimensional extremally disconnected frames.