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Topological Representation of Contact Lattices

  • Ivo Düntsch
  • Wendy MacCaull
  • Dimiter Vakarelov
  • Michael Winter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4136)

Abstract

The theory of Boolean contact algebras has been used to represent a region based theory of space. Some of the primitives of Boolean algebras are not well motivated in that context. One possible generalization is to drop the notion of complement, thereby weakening the algebraic structure from Boolean algebra to distributive lattice. The main goal of this paper is to investigate the representation theory of that weaker notion, i.e., whether it is still possible to represent each abstract algebra by a substructure of the regular closed sets of a suitable topological space with the standard Whiteheadean contact relation.

Keywords

Topological Space Boolean Algebra Distributive Lattice Contact Structure Topological Representation 
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. 1.
    Allwein, G., MacCaull, W.: A Kripke semantics for the logic of Gelfand quantales. Studia Logica 61, 1–56 (2001)MathSciNetGoogle Scholar
  2. 2.
    Balbes, R., Dwinger, P.: Distributive Lattices. University of Missouri Press, Columbia (1974)Google Scholar
  3. 3.
    Cornish, W.H.: Crawley’s completion of a conditionally upper continuous lattice. Pac. J. Math. 51(2), 397–405 (1974)MathSciNetMATHGoogle Scholar
  4. 4.
    Dimov, G., Vakarelov, D.: Contact algebras and region–based theory of space: A proximity approach. Fundamenta Informaticae (2006) (to appear)Google Scholar
  5. 5.
    Dimov, G., Vakarelov, D.: Topological Representation of Precontact algebras. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 1–16. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Düntsch, I., Vakarelov, D.: Region–based theory of discrete spaces: A proximity approach. Discrete Applied Mathematics (2006) (to appear)Google Scholar
  7. 7.
    Düntsch, I., Winter, M.: Lattices of contact relations (2005) (Preprint)Google Scholar
  8. 8.
    Düntsch, I., Winter, M.: A representation theorem for Boolean contact algebras. Theoretical Computer Science (B) 347, 498–512 (2005)CrossRefMATHGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Engelking, R.: General topology, PWN (1977)Google Scholar
  11. 11.
    MacCaull, W., Vakarelov, D.: Lattice-based paraconsistent logic. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 173–187. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Stone, M.: Topological representations of distributive lattices and Brouwerian logics. Casopis Pest. Mat. 67, 1–25 (1937)Google Scholar
  13. 13.
    Vakarelov, D., Düntsch, I., Bennett, B.: A note on proximity spaces and connection based mereology. In: Welty, C., Smith, B. (eds.) Proceedings of the 2nd International Conference on Formal Ontology in Information Systems (FOIS 2001), pp. 139–150. ACM, New York (2001)CrossRefGoogle Scholar
  14. 14.
    Vakarelov, D., Dimov, G., Düntsch, I., Bennett, B.: A proximity approach to some region-based theory of space. Journal of applied non-classical logics 12(3-4), 527–559 (2002)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Wallman, H.: Lattices and topological spaces. Math. Ann. 39, 112–136 (1938)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ivo Düntsch
    • 1
  • Wendy MacCaull
    • 2
  • Dimiter Vakarelov
    • 3
  • Michael Winter
    • 1
  1. 1.Department of Computer ScienceBrock UniversitySt. CatharinesCanada
  2. 2.Department of Mathematics, Statistics and Computer ScienceSt. Francis Xavier UniversityAntigonishCanada
  3. 3.Department of Mathematical LogicSofia UniversitySofiaBulgaria

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