In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to...

The dimension of a partiruly ordered set (X, P) is the smallest positive integer i for which there exists a function f which assigns to each x E X a sequence {f(x)(i): I s i 41) of TeaI numbers so that x Sy in P i! and only if f(x)(i)Sf(y)(i) for each i = 1,2,. , , , t. TIE irltelval ~lmension of (X, P) is the sm;.llcst integer t for which there exists a function F which assigns to 2ach x E X a...

In this article we describe a very efficient method to construct pairwise nonisomorphic posets (equivalently, T0 topologies). We also give the results obtained by a computer program based on this algorithm, in particular the numbers of nonisomorphic posets on 15 and 16 points and the numbers of labelled posets and topologies on 17 and 18 points.

In every dense poset P every maximal antichain S may be partitioned into disjoint subsets S1 and S2 , such that the union of the upset of S1 with the downset of S2 yields the entire poset: U(S1) [ D(S2) = P . To nd a similar splitting of maximal antichains in posets is NP{hard in general.

We study a class of algebras associated with linear spaces and its relations with polymatroids and integral posets, i.e. posets supporting homogeneous ASL. We prove that the base ring of a transversal polymatroid is Koszul and describe a new class of integral posets. As a corollary we obtain that every Veronese subring of a polynomial ring is an ASL.

We show that the well known homotopy complementation formula of Björner andWalker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, G̃n(R) and expn(X) which were introduced and studied by V. Vassiliev. Among o...

It has been known for 30 years that posets with bounded height and with cover graphs of bounded maximum degree have bounded dimension. Recently, Streib and Trotter proved that dimension is bounded for posets with bounded height and planar cover graphs, and Joret et al. proved that dimension is bounded for posets with bounded height and with cover graphs of bounded tree-width. In this paper, it ...

In this paper we give convex-ear decompositions for the order complexes of several classes of posets, namely supersolvable lattices with non-zero Möbius functions and rank-selected subposets of such lattices, rank-selected geometric lattices, and rank-selected face posets of shellable complexes which do not include the top rank. These decompositions give us many new inequalities for the h-vecto...

Let P be a finite poset. We consider the hypergraph H(P ) whose vertices are the elements of P and whose edges are the maximal intervals of P . It is known that H(P ) has the König and dual König properties for the class of series-parallel posets. Here we introduce a new class which contains series-parallel posets and for which the dual König property is satisfied. For the class of N-free poset...