Height two orders and convex families

Given a partially ordered set the convex sets of the partial order form an atomistic convex geometry and the height two partial orders form a subclass closed family. These two families correspond to one another as the minimal family that generates a convex family under the convex closure operation is always going to be a height two partial order. Consider the following height two family:

#{#{0} #{1} #{2} #{3} #{0 1 2} #{1 2 3}}

The convex closure of the above height two family is a family which contains all of the elements of the height two family plus those elements that are necessary to make the family convex:

#{#{0} #{1} #{2} #{3} #{0 1} #{0 2} 
  #{1 2} #{1 3} #{2 3} #{0 1 2} #{0 1 3}}

Certain partial orders can be both height two and convex. These include the dependency families which are the subsingleton closed nullfree rank two families. The dependency families correspond to the simple graphs in graph theory and they can also be produced from any atomistic height two order in which no element has more then two predecessors.

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.