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Starting with the definition of an incidence algebra associated to a partially ordered set, we discuss various examples including the incidence algebra of formal power series, of exponential power series, of Eulerian power series, and of formal Dirichlet series. We then use the algebra-coalgebra duality to realize these examples as dual algebras of certain coalgebras. Finally, we use this fact to identify their algebra Hochschild cohomology groups with the Cotor-groups of the dual coalgebras. (Joint work with M. Kanuni Er & A. Kaygun)