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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bloch, I., Bretto, A.: Mathematical Morphology on Hypergraphs: Preliminary Definitions and Results. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 429–440. Springer, Heidelberg (2011)
Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press (1974)
Bloch, I., Heijmans, H.J.A.M., Ronse, C.: Mathematical morphology. In: Aiello, M., Pratt-Hartmann, I., van Benthem, J. (eds.) Handbook of Spatial Logics, ch. 14, pp. 857–944. Springer (2007)
Brown, R., Morris, I., Shrimpton, J., Wensley, C.D.: Graphs of Morphisms of Graphs. Bangor Mathematics Preprint 06.04, Mathematics Department, University of Wales, Bangor (2006)
Borceux, F.: Handbook of Categorical Algebra. Basic Category Theory, vol. 1. Cambridge University Press (1994)
Cousty, J., Najman, L., Serra, J.: Some Morphological Operators in Graph Spaces. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) ISMM 2009. LNCS, vol. 5720, pp. 149–160. Springer, Heidelberg (2009)
Heijmans, H., Vincent, L.: Graph Morphology in Image Analysis. In: Dougherty, E.R. (ed.) Mathematical Morphology in Image Processing, ch. 6, pp. 171–203. Marcel Dekker (1993)
Lawvere, F.W.: Introduction. In: Lawvere, F.W., Schanuel, S.H. (eds.) Categories in Continuum Physics. Lecture Notes in Mathematics, vol. 1174, pp. 1–16. Springer (1986)
Lawvere, F.W.: Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In: Carboni, A., et al. (eds.) Category Theory, Proceedings, Como 1990. Lecture Notes in Mathematics, vol. 1488, pp. 279–281. Springer (1991)
Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. A First Introduction to Topos Theory. Springer (1992)
Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)
Rosenthal, K.I.: Quantales and their applications. Pitman Research Notes in Mathematics, vol. 234. Longman (1990)
Schmidt, G.: Relational Mathematics. Encyclopedia of Mathematics and its Applications, vol. 132. Cambridge University Press (2011)
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)
Stell, J.G.: Relations in Mathematical Morphology with Applications to Graphs and Rough Sets. In: Winter, S., Duckham, M., Kulik, L., Kuipers, B. (eds.) COSIT 2007. LNCS, vol. 4736, pp. 438–454. Springer, Heidelberg (2007)
Stell, J.G.: Relational Granularity for Hypergraphs. In: Szczuka, M., Kryszkiewicz, M., Ramanna, S., Jensen, R., Hu, Q. (eds.) RSCTC 2010. LNCS (LNAI), vol. 6086, pp. 267–276. Springer, Heidelberg (2010)
Taylor, P.: Practical Foundations of Mathematics. Cambridge University Press (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Stell, J.G. (2012). Relations on Hypergraphs. In: Kahl, W., Griffin, T.G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2012. Lecture Notes in Computer Science, vol 7560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33314-9_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-33314-9_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33313-2
Online ISBN: 978-3-642-33314-9
eBook Packages: Computer ScienceComputer Science (R0)