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?
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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.
answered Jun 18, 2021 at 20:05
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$\begingroup$ Oh thank you very much! This makes a certain proof a lot simpler than I though. $\endgroup$ Commented Jun 21, 2021 at 13:05