Skip to main content
Springer Nature Link
Log in

The algebraic structure of sets of regions

  • Topological Models of Space
  • Conference paper
  • First Online:
Spatial Information Theory A Theoretical Basis for GIS (COSIT 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1329))

Included in the following conference series:

  • 291 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. 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. 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. 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. B. L. Clarke. A calculus of individuals based on ‘connection'. Notre Dame Journal of Formal Logic, 22:204–218, 1981.

    Google Scholar 

  5. R. Franzosa and M. J. Egenhofer. Point-set topological spatial relations. International Journal of Geographical Information Systems, 5:161–174, 1991.

    Google Scholar 

  6. 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. 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. 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. P. T. Johnstone. Stone Spaces. Cambridge University Press, 1982.

    Google Scholar 

  10. M. Johnson. The Body in the Mind: The Bodily Basis of Meaning, Imagination and Reason. University of Chicago Press, 1987.

    Google Scholar 

  11. 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. 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. 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. G. E. Reyes and H. Zolfaghari. Bi-Heyting algebras, toposes and modalities. Journal of Philosophical Logic, 25:25–43, 1996.

    Google Scholar 

  15. B. Smith. Mereotopology — A theory of parts and boundaries. Data and Knowledge Engineering, 20:287–303, 1996.

    Google Scholar 

  16. J. G. Stell. A lattice theoretic account of spatial regions. Technical report, Keele University, Department of Computer Science, 1997.

    Google Scholar 

  17. S. Vickers. Topology via Logic. Cambridge University Press, 1989. *** DIRECT SUPPORT *** A0008169 00004

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Keele University, ST5 5BG, Keele, Staffs, UK

    J. G. Stell & M. F. Worboys

Authors
  1. J. G. Stell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. F. Worboys
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Stephen C. Hirtle Andrew U. Frank

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Stell, J.G., Worboys, M.F. (1997). The algebraic structure of sets of regions. In: Hirtle, S.C., Frank, A.U. (eds) Spatial Information Theory A Theoretical Basis for GIS. COSIT 1997. Lecture Notes in Computer Science, vol 1329. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63623-4_49

Download citation

  • DOI: https://doi.org/10.1007/3-540-63623-4_49

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63623-6

  • Online ISBN: 978-3-540-69616-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics