2
$\begingroup$

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?

$\endgroup$

1 Answer 1

5
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Oh thank you very much! This makes a certain proof a lot simpler than I though. $\endgroup$ Jun 21, 2021 at 13:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.