Hypergraph rewriting using conformisms

In this paper we study single-pushout transformation in a category of spans, a generalization of the notion of partial morphism in, for instance, 2,4]. As an application, single-pushout transformation in a category of hypergraphs with a special type of partial morphisms, the conformisms, is presented. In particular , we show the existence of the pushout of any pair of conformisms of hypergraphs with the same source hypergraph, and how to construct one such a pushout. Finally, hypergraph rewriting using conformisms is compared to single-pushout hypergraph rewriting by means of a detailed example. We present in this paper an approach to hypergraph rewriting by means of single-pushout transformation in a category of hypergraphs with a special type of partial morphisms, the conformisms, inspired by the homonymous notion in the theory of partial algebras. Although this approach has already been treated by some of us in 1] by means of what we could call \rude-force methods," in this paper we develop it as a special case of single-pushout transformation in a category of spans, a generalization of the notion of partial morphism in, for instance, 2,4]. So, the main result in this note establishes a necessary condition for the existence of the pushout of two such spans. This condition has a part involving properties of the \original" category, from which the category of spans is derived, and a part referring to the speciic spans. Then, we show that such a necessary condition is always satissed in the case of pairs of conformisms of hypergraphs, understood as spans in a category of hypergraphs with \weak" morphisms. This type of arguments using suitable categories of (sort of) partial mor-phisms is well-known, and in particular it was explained to us by M. LL owe (whom we deeply acknowledge with thanks for it). The novelty in our approach lies in part on the fact that our partial morphisms are not equivalence classes as in 2] but single objects, as it is usual to think about them, on the other hand. c 1995 Elsevier Science B. V.