Relations on hypergraphs

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Abstract

A relation on a hypergraph is a binary relation on the set consisting of the nodes and hyperedges together, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join-preserving mappings on the lattice of sub-hypergraphs. This paper studies the algebra of these relations, in particular the analogues of the familiar operations of complement and converse of relations. When generalizing from relations on a set to relations on a hypergraph we find that the Boolean algebra of relations is replaced by a weaker structure: a pair of isomorphic bi-Heyting algebras, one of which arises from the relations on the dual hypergraph. The paper also considers the representation of sub-hypergraphs as relations and applies the results obtained to mathematical morphology for hypergraphs. © 2012 Springer-Verlag.

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Stell, J. G. (2012). Relations on hypergraphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7560 LNCS, pp. 326–341). https://doi.org/10.1007/978-3-642-33314-9_22

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