Planar lattices are lexicographically shellable

The special properties of planar posets have been studied, particularly in the 1970's by I. Rival and others. More recently, the connection between posets, their corresponding polynomial rings and corresponding simplicial complexes has been studied by Stanley and others. This paper, using work of Björner, provides a connection between the two bodies of work, by characterizing when planar posets are Cohen-Macaulay. Planar posets are lattices when they contain a greatest and a least element. We show that a finite planar lattice is lexicographically shellable and therefore Cohen-Macaulay iff it is rank-connected.