Posets That Locally Resemble Distributive Lattices: An Extension of Stanley's Theorem

Jonathan David Farley and Stefan E. Schmidt

Let $P$ be a graded poset with 0 and 1 and rank at least 3. Assume that every rank 3 interval is a distributive lattice and that, for every interval of rank at least 4, the interval minus its endpoints is connected. It is shown that $P$ is a distributive lattice, thus resolving an issue raised by Stanley. Similar theorems are proven for semimodular, modular, and complemented modular lattices.

As a corollary, a theorem of Stanley for Boolean lattices is obtained, as well as a theorem of Grabiner (conjectured by Stanley) for products of chains. Applications to geometry and connections with the theory of buildings are discussed.