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The algebraic structure of sets of regions

  • J. G. Stell
  • M. F. Worboys
Topological Models of Space
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1329)

Abstract

The provision of ontologies for spatial entities is an important topic in spatial information theory. Heyting algebras, co-Heyting algebras, and bi-Heyting algebras are structures having considerable potential for the theoretical basis of these ontologies. This paper gives an introduction to these Heyting structures, and provides evidence of their importance as algebraic theories of sets of regions. The main evidence is a proof that elements of certain Heyting algebras provide models of the Region-Connection Calculus developed by Cohn et al. By using the mathematically well known techniques of “pointless topology”, it is straight-forward to conduct this proof without any need to assume that regions consist of sets of points. Further evidence is provided by a new qualitative theory of regions with indeterminate boundaries. This theory uses modal operators which are related to the algebraic operations present in a bi-Heyting algebra.

Keywords

Topological Space Boolean Algebra Proper Part Heyting Algebra Spatial Entity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [BF96]
    P. A. Burrough and A. U. Frank, editors. Geographic Objects with Indeterminate Boundaries, volume 2 of GISDATA Series. Taylor and Francis, 1996.Google Scholar
  2. [CdF96]
    E. Clementini and P. di Felici. An algebraic model for spatial objects with indeterminate boundaries. In Burrough and Frank [BF96], pages 155–169.Google Scholar
  3. [CG96]
    A. G. Cohn and N. M. Gotts. The ‘egg-yolk’ representation of regions with indeterminate boundaries. In Burrough and Frank [BF96], pages 171–187.Google Scholar
  4. [Cla81]
    B. L. Clarke. A calculus of individuals based on ‘connection'. Notre Dame Journal of Formal Logic, 22:204–218, 1981.Google Scholar
  5. [FE91]
    R. Franzosa and M. J. Egenhofer. Point-set topological spatial relations. International Journal of Geographical Information Systems, 5:161–174, 1991.Google Scholar
  6. [FE95]
    R. Franzosa and M. J. Egenhofer. On the equivalence of topological relations. International Journal of Geographical Information Systems, 9:133–152, 1995.Google Scholar
  7. [GGC96]
    N. M. Gotts, J. M. Gooday, and A. G. Cohn. A connection based approach to common-sense topological description and reasoning. The Monist, 79:51–75, 1996.Google Scholar
  8. [Got96]
    N. M. Gotts. An axiomatic approach to topology for spatial information systems. Research Report 96.25, University of Leeds, School of Computer Studies, 1996.Google Scholar
  9. [Joh82]
    P. T. Johnstone. Stone Spaces. Cambridge University Press, 1982.Google Scholar
  10. [Joh87]
    M. Johnson. The Body in the Mind: The Bodily Basis of Meaning, Imagination and Reason. University of Chicago Press, 1987.Google Scholar
  11. [Law86]
    F. W. Lawvere. Introduction. In F. W. Lawvere and S. H. Schanuel, editors, Categories in Continuum Physics, volume 1174 of Lecture Notes in Mathematics, pages 1–16. Springer-Verlag, 1986.Google Scholar
  12. [Law91]
    F. W. Lawvere. Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In A. Carboni et al., editors, Category Theory, Proceedings, Como 1990, volume 1488 of Lecture Notes in Mathematics, pages 279–281. Springer-Verlag, 1991.Google Scholar
  13. [MR94]
    F. Magnan and G. E. Reyes. Category theory as a conceptual tool in the study of cognition. In J. Macnamara and G. E. Reyes, editors, The Logical Foundations of Cognition, chapter 5, pages 57–90. Oxford University Press, 1994.Google Scholar
  14. [RZ96]
    G. E. Reyes and H. Zolfaghari. Bi-Heyting algebras, toposes and modalities. Journal of Philosophical Logic, 25:25–43, 1996.Google Scholar
  15. [Smi96]
    B. Smith. Mereotopology — A theory of parts and boundaries. Data and Knowledge Engineering, 20:287–303, 1996.Google Scholar
  16. [Ste97]
    J. G. Stell. A lattice theoretic account of spatial regions. Technical report, Keele University, Department of Computer Science, 1997.Google Scholar
  17. [Vic89]
    S. Vickers. Topology via Logic. Cambridge University Press, 1989. *** DIRECT SUPPORT *** A0008169 00004Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • J. G. Stell
    • 1
  • M. F. Worboys
    • 1
  1. 1.Department of Computer ScienceKeele UniversityKeele, StaffsUK

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