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International Journal of Theoretical Physics

, Volume 44, Issue 12, pp 2283–2293 | Cite as

The Canonical Topology on a Meet-Semilattice

  • Isar Stubbe
Article

Abstract

Considering the lattice of properties of a physical system, it has been argued elsewhere that—to build a calculus of propositions having a well-behaved notion of disjunction (and implication)—one should consider a very particular frame completion of this lattice. We show that the pertinent frame completion is obtained as sheafification of the presheaves on the given meet-semilattice with respect to its canonical Grothendieck topology, an explicit description of which is easily given. Our conclusion is that there is an intrinsic categorical quality to the notion of “disjunction” in the context of property lattices of physical systems.

Keywords

meet-semilattice topology sheaf frame completion 

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References

  1. Aerts, D. (1982). Description of many separated physical entities without the paradoxes encountered in quantum mechanics. Foundations of Physics 12, 1131–1170.CrossRefMathSciNetGoogle Scholar
  2. Borceux, F. (1994). Handbook of Categorical Algebra, Cambridge University Press, Cambridge, UK.Google Scholar
  3. Borceux, F. and Quinteiro, C. (1996). A theory of enriched sheaves. Cahiers de Topologie et Géométrie Différentielle. Catégoriques 37, 145–162.MathSciNetGoogle Scholar
  4. Borceux, F. and Stubbe, I. (2000). Short introduction to enriched categories. Fundamental Theories of Physics 111. Current Research in Operational Quantum Logic: Algebras, Categories, Languages, Kluwer Academic, Dordrecht, The Netherlands, pp. 167–194.Google Scholar
  5. Bruns, G. and Lakser, H. (1970). Injective hulls of semilattices. Canadian Mathematical Bulletin 13, 115–118.MathSciNetGoogle Scholar
  6. Coecke, B. (2001). Quantum logic in intuitionistic perspective. Studia Logica 70, 411–440.CrossRefMathSciNetGoogle Scholar
  7. Coecke, B., Moore, D., and Stubbe, I. (2001). Quantaloids describing causation and propagation for physical entities. Foundations of Physics Letters 14, 133–145.CrossRefMathSciNetGoogle Scholar
  8. Coecke, B. and Stubbe, I. (1999a). On a duality of quantales emerging from an operational resolution. International Journal of Theoretical Physics 38, 3269–3281.CrossRefMathSciNetGoogle Scholar
  9. Coecke, B. and Stubbe, I. (1999b). Operational resolution and state transitions in a categorical setting. Foundations of Physics Letters 12, 29–49.CrossRefMathSciNetGoogle Scholar
  10. Dowker, C. H. and Papert, D. (1966). Quotient frames and subspaces. Proceedings of the London Mathematical Society 16, 275–296.Google Scholar
  11. Johnstone, P. T. (1982). Stone Spaces, Cambridge University Press, Cambridge, UK.MATHGoogle Scholar
  12. MacLane, S. (1971). Categories for the Working Mathematician, Springer, Berlin.Google Scholar
  13. MacLane, S. and Moerdijk, I. (1992). Sheaves in Geometry and Logic, Springer, Berlin.Google Scholar
  14. Moore, D. (1995). Categories of representations of physical systems. Helvetica Physica Acta 68, 658–678.MathSciNetGoogle Scholar
  15. Piron, C. (1972). Survey of general quantum physics. Foundations of Physics 2, 287–314.CrossRefMathSciNetGoogle Scholar
  16. Piron, C. (1976). Foundations of Quantum Physics, Benjamin, New York.MATHGoogle Scholar
  17. Piron, C. (1990). Mécanique quantique. Bases et applications, Presses Polytechniques et Universitaires Romandes, Lausanne.Google Scholar
  18. Stubbe, I. (2001). Logique quantique opérationelle et catégories, Compte-rendu du Séminaire Itinérant de Catégories, LAMFA, Université de Picardie-Jules Verne Amiens, Vol. 18, pp. 3–14.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité de LouvainLouvain-la-NeuveBelgium

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