Hopf Algebra of Building Sets

The combinatorial Hopf algebra on building sets BSet extends the chromatic Hopf algebra of simple graphs. The image of a building set under canonical morphism to quasi-symmetric functions is the chromatic symmetric function of the corresponding hypergraph. By passing from graphs to building sets, we construct a sequence of symmetric functions associated to a graph. From the generalized DehnSommerville relations for the Hopf algebra BSet, we define a class of building sets called eulerian and show that eulerian building sets satisfy Bayer-Billera relations. We show the existence of the cd−index, the polynomial in two noncommutative variables associated to an eulerian building set. The complete characterization of eulerian building sets is given in terms of combinatorics of intersection posets of antichains of finite sets.