Rewriting in categories of spans

Single-pushout transformation in a category of spans, in some sense a generalization of the usual notion of partial morphism, is studied in this paper. Contrary to the usual notion of partial morphism, spans are single objects instead of equivalence classes. A necessary condition for the existence of the pushout of two spans is established which involves properties of the base category, from which the category of spans is derived, as well as properties of the spans themselves. Several interesting categories of partial morphisms of hypergraphs are proved to satisfy the necessary condition