A (possibly open) problem about 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.

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