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 R. Stanley and others. This paper, using work of A. Bjorner, 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 nite planar lattice is lexicographically shellable and therefore Cohen-Macaulay i it is rankconnected. Every poset corresponds to a simplicial complex, called the order complex, de ned by taking chains of the poset to be faces of the simplicial complex. We de ne a face of a simplicial complex to include both interior and boundary. A pure, nite simplicial complex is shellable if its maximal faces (facets) can be ordered F1; F2; : : : ; Fn in such a way that Fk\([ k 1 i=1 Fi) is a nonempty union of maximal proper faces of Fk for 2 k n. Shellable triangulations of spheres and balls are discussed in [7]. A poset is said to be shellable if its order complex is shellable. Let k be a eld. Then with every poset P there is also associated a polynomial ring k(P ) whose variables correspond to the vertices of the poset, in which a product of two or more variables is zero if and only if it contains an independent set of vertices of the poset. Thus it is possible to ask if this ring is Cohen-Macaulay (C-M); i.e., if the depth of k(P ) is equal to the (Krull) dimension of k(P ). See [19] for background. (The Krull dimension of a polynomial ring is di erent from the combinatorial de nition of the dimension of a poset. See de nition below.) Let f be a face in a simplicial complex C, and link(f) be the subcomplex of C consisting of the set of faces g in C such that f [ g is a face in C and f \g = ;. The topological dimension of a simplicial complex is de ned to be one less than the size of its largest face. Then a simplicial complex is de ned to be C-M when for every face f in the complex C the reduced homology