On the flag f-vector of a graded lattice with nontrivial homology

TL;DR

This paper proves that the Boolean algebra minimizes the flag f-vector among certain graded lattices with nontrivial top homology, and conjectures a similar property for the flag h-vector in Cohen-Macaulay cases.

Contribution

It establishes a minimality property of the Boolean algebra for the flag f-vector in graded lattices with nontrivial homology and proposes a related conjecture for the flag h-vector.

Findings

Boolean algebra minimizes flag f-vector in specified lattices

Conjecture on flag h-vector minimality in Cohen-Macaulay lattices

Provides a new extremal property related to lattice topology

Abstract

It is proved that the Boolean algebra of rank n minimizes the flag f-vector among all graded lattices of rank n, whose proper part has nontrivial top-dimensional homology. The analogous statement for the flag h-vector is conjectured in the Cohen-Macaulay case.