Posets that locally resemble distributive lattices - An extension of Stanley's theorem (with connections to buildings and diagram geometries)

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 Stanle y. Similar theorems are proven for semi modular, 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 incidence geometry and connections wit h the theory of buildings are discussed. (C) 2000 Academic Press.