Skip to main content
SpringerLink
Log in

On the Skeleton of Stonian p-Ortholattices

  • Conference paper
Relations and Kleene Algebra in Computer Science (RelMiCS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5827))

Included in the following conference series:

Abstract

Boolean Contact Algebras (BCA) establish the algebraic counterpart of the mereotopolopy induced by the Region Connection Calculus (RCC). Similarly, Stonian p-ortholattices serve as a lattice theoretic version of the ontology RT − of Asher and Vieu. In this paper we study the relationship between BCAs and Stonian p-ortholattices. We show that the skeleton of every Stonian p-ortholattice is a BCA, and, conversely, that every BCA is isomorphic to the skeleton of a Stonian p-ortholattice. Furthermore, we prove the equivalence between algebraic conditions on Stonian p-ortholattices and the axioms C5, C6, and C7 for BCAs.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asher, N., Vieu, L.: Toward a geometry of common sense: a semantics and a complete axiomatization for mereotopology. In: Proc. of the 14th Int. Joint Conf. on Artificial Intelligence (IJCAI 1995), pp. 846–852. Morgan Kaufmann, San Francisco (1995)

    Google Scholar 

  2. Clarke, B.: A calculus of individuals based on ‘Connection’. Notre Dame Journal of Formal Logic 22(3), 204–218 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cohn, A.G., Bennett, B., Gooday, J.M., Gotts, N.M.: RCC: a calculus for region based qualitative spatial reasoning. GeoInformatica 1, 275–316 (1997)

    Article  Google Scholar 

  4. Cohn, A.G., Renz, J.: Qualitative spatial representation and reasoning. In: Handbook of Knowledge Representation. Elsevier, Amsterdam (2008)

    Google Scholar 

  5. Dimov, G., Vakarelov, D.: Contact Algebras and Region-based Theory of Space: A Proximity Approach. Fundamenta Informaticae 74(2-3), 209–282 (2006)

    MATH  MathSciNet  Google Scholar 

  6. Düntsch, I., Winter, M.: A representation theorem for boolean contact algebras. Theoretical Computer Science 347, 498–512 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Düntsch, I., Winter, M.: Weak contact structures. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 73–82. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  8. Frink, O.: Pseudo-Complements in Semi-Lattices. Duke Mathematical Journal 29(4), 505–514 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  9. Glivenko, V.: Sur quelque points de la logique de M. Brouwer. Bulletin de la Classe des Sciences, 5e série, Académie royale de Belgique 15, 183–188 (1929)

    Google Scholar 

  10. Hahmann, T., Winter, M., Gruninger, M.: Stonian p-ortholattices: A new approach to the mereotopology RT0. Artificial Intelligence (May 2009) (accepted)

    Google Scholar 

  11. Stell, J.G.: Boolean connection algebras: a new approach to the Region-Connection Calculus. Artificial Intelligence 122, 111–136 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Vakarelov, D., Dimov, G., Düntsch, I., Bennett, B.: A proximity approach to some region-based theories of space. Journal of Applied Non-classical Logics 12(3-4), 527–559 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

  2. Department of Computer Science, University of Toronto,  

    Torsten Hahmann

  3. Department of Mechanical and Industrial Engineering, Department of Computer Science, University of Toronto,  

    Michael Gruninger

Authors
  1. Michael Winter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Torsten Hahmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Gruninger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Rudolf Berghammer

  2. College of Engineering, Qatar University, P.O. Box 2713, CSE 209, Doha, Qatar,  

    Ali Mohamed Jaoua

  3. Institut für Informatik, Universität Augsburg, Universitätsstr. 6a, 86135, Augsburg, Germany

    Bernhard Möller

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Winter, M., Hahmann, T., Gruninger, M. (2009). On the Skeleton of Stonian p-Ortholattices. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2009. Lecture Notes in Computer Science, vol 5827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04639-1_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04639-1_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04638-4

  • Online ISBN: 978-3-642-04639-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation