Construction of Pushout Complements in the Category of Hypergraphs

We describe a concrete construction of all pushout complements for two given morphisms f : A -> B, m: B -> D in the category of hypergraphs, valid also for the case where f, m are non-injective. It is based on the generation of suitable equivalence relations. We also give a combinatorial interpretation and show how well-known coefficients from combinatorics, such as the Bell numbers, can be recovered. Furthermore we present a formula that can be used to compute the number of pushout complements for two given morphisms.