Antipodes and incidence coalgebras

We introduce the notion of a reduced incidence coalgebra of a family of locally finite partially ordered sets and describe how bialgebras and Hopf algebras arise as such coalgebras. In the case when a reduced incidence coalgebra is a Hopf algebra, we give explicit recursive and closed formulas for the antipode, each of which generalizes a corresponding formula from the theory of Möbius functions. We give applications to the inversion of ordinary formal power series and inversion of Dirichlet series. We also formulate the classical Lagrange inversion formula as a combinatorial formula for the antipode of a Hopf algebra arising from the family of finite partition lattices.