In this paper we introduce a class of partially ordered sets, called differential posets, with many remarkable combinatorial and algebraic properties. (Terminology from lattice theory and the theory of partially ordered sets not explained here may be found in [St6].) The combinatorial properties of differential posets are concerned with the counting of saturated chains Xl < x2 < ... < x k or of...

For partially ordered sets that are continuous in the sense of D. S. Scott, the waybelow relation is crucial. It expresses the approximation of an ideal element by its finite parts. We present explicit characterizations of the way-below relation on spaces of continuous functions from topological spaces into continuous posets. Although it is well-known in which cases these function spaces are co...

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D(X) and D(Y ) are equivalent as triangulated categories. We give explicit combinatorial properties of X which are invariant under derived equivalence, ...

Using countable support iterations of S -proper posets, we show that the existence of a ∆3 definable wellorder of the reals is consistent with each of the following: d < c, b < a = s, b < g.

Results of R. Stanley and M. Masuda completely characterize the hvectors of simplicial posets whose order complexes are spheres. In this paper we examine the corresponding question in the case where the order complex is a ball. Using the face rings of these posets, we develop a series of new conditions on their h-vectors. We also present new methods for constructing poset balls with specific h-...

In this paper, we consider upper bound graphs and double bound graphs of posets. We obtain a characterization of upper bound graphs whose complements are also upper bound graphs as follows: for a connected graph G, both G and G are upper bound graphs if and only if G is a split graph with V(G)= K + S, where K is a clique and S is an independent set, satisfying one of the following conditions: (...

Macaulay posets are posets for which there is an analogue of the classical KruskalKatona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets. Emphasis is also put on construction of extremal ideals in Macaulay posets.