For a convex polytope, its face poset is combinatorially determined by vertexfacet 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 vertexfacet incidences to still store all the information? Is it the property of being an abstract polytope, or will something weaker be enough?
1 Answer
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If $L$ is a finite lattice, then $L$ is determined by its subposet of elements that are joinirreducible or meetirreducible (or both). In particular, if the only joinirreducibles are atoms (vertices) and only meetirreducibles 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.

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