Posets That Locally Resemble Distributive Lattices: An Extension of Stanley's Theorem (with Connections to Buildings and Diagram Geometries)
✍ Scribed by Jonathan David Farley; Stefan E. Schmidt
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 335 KB
- Volume
- 92
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
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 incidence geometry and connections with the theory of buildings are discussed.