On posets and hopf algebras

We generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in terms Hopf algebras. In the last sections of the paper we generalize the reduced incidence Hopf algebra of posets to the Hopf algebra of hierarchical simplicial complexes.