Independent sets of Moore families

Lets suppose that we have a family of sets that forms a Moore family.

#{#{} #{0} #{1} #{2} #{0 1 2}}

Then the independent sets of that Moore family are precisely those sets that form power sets when intersected with all the sets of the Moore family. Here are the independent sets of the above Moore family:

#{#{} #{0} #{1} #{2}}

If a given Moore family is a family of flats formed from a matroid then the independent sets of that family can be used to reproduce the matroid. If the Moore family is instead an Alexandrov family then the independent sets are the cliques of the complement of the comparability relation of the specialization preorder. This means that the independent sets of an Alexandrov family form a clique complex.