The algebra of set functions I: The product theorem and duality.
Résumé
We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.