For a convex polytope, its face poset is combinatorially determined by vertex-facet incidences. Now suppose we have an arbitrary finite poset that is *ranked*, so I can still speak of vertices and facets. What property should be satisfied for vertex-facet incidences to still store all the information? Is it the property of being an abstract polytope, or will something weaker be enough?



What property of ranked poset ensures that it is determined by its vertex-facet incidences?
If $L$ is a finite lattice, then $L$ is determined by its subposet of elements that are join-irreducible or meet-irreducible (or both). In particular, if the only join-irreducibles are atoms (vertices) and only meet-irreducibles are coatoms (facets), then $L$ is determined by the incidences between its vertices and facets. See for instance Exercise 3.27 of *Enumerative Combinatorics*, vol. 1, second edition.


