6533b827fe1ef96bd12853fc

RESEARCH PRODUCT

Posets That Locally Resemble Distributive Lattices

Jonathan David FarleyJonathan David FarleyStefan E. SchmidtStefan E. SchmidtStefan E. Schmidt

subject

Modular latticeDiscrete mathematicsDistributive latticeCongruence lattice problemMap of latticesTheoretical Computer ScienceComplemented latticeCombinatoricsGraded posetComputational Theory and MathematicsSemimodular latticeDiscrete Mathematics and CombinatoricsBirkhoff's representation theoremMathematics

description

Abstract 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.

yearjournalcountryeditionlanguage
2000-11-01Journal of Combinatorial Theory, Series A
https://doi.org/10.1006/jcta.1999.3049