Relational Granularity for Hypergraphs

A set of subsets of a set may be seen as granules that allow arbitrary subsets to be approximated in terms of these granules. In the simplest case of rough set theory, the set of granules is required to partition the underlying set, but granulations based on relations more general than equivalence relations are well-known within rough set theory. The operations of dilation and erosion from mathematical morphology, together with their converse forms, can be used to organize different techniques of granular approximation for subsets of a set with respect to an arbitrary relation. The extension of this approach to granulations of sets with structure is examined here for the case of hypergraphs. A novel notion of relation on a hypergraph is presented, and the application of these relations to a theory of granularity for hypergraphs is discussed.