Applied Categorical Structures

, Volume 10, Issue 5, pp 469–480 | Cite as

On the Amnestic Modification of the Category of State Property Systems

  • D. Aerts
  • E. Colebunders
  • A. Van der Voorde
  • B. Van Steirteghem
Article

Abstract

State property systems were created on the basis of physical intuition in order to describe a mathematical model for physical systems. A state property system consists of a triple: a set of states, a complete lattice of properties and a specified function linking the other two components. The definition of morphisms between such objects was inspired by the physical idea of a subsystem. In this paper we give an isomorphic description of the category of state property systems, thus introducing a category SP, which is concrete over Set. This isomorphic description enables us to investigate further categorical properties of SP. It turns out that the category SP is not amnestic. In our main theorem we prove that the amnestic modification of SP is the construct Cls of closure spaces and continuous maps. Moreover we observe that the categorical product and coproduct in Cls find, through SP, application in physics.

closure space state property system amnestic modification 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adámek, J., Herrlich, H. and Strecker, G. E.: Abstract and Concrete Categories, Wiley, New York, 1990.Google Scholar
  2. 2.
    Aerts, D.: Description of many physical entities without the paradoxes encountered in quantum mechanics, Foundations of Physics 12 (1982), 1131-1170.Google Scholar
  3. 3.
    Aerts, D.: Construction of the tensor product for the lattices of properties of physical entities, Journal of Mathematical Physics 25 (1984), 1434-1441.Google Scholar
  4. 4.
    Aerts, D.: Quantum structures, separated physical entities and probability, Foundations of Physics 24 (1994), 1227-1259.Google Scholar
  5. 5.
    Aerts, D.: Foundations of quantum physics: a general realistic and operational approach, International Journal of Theoretical Physics 38 (1999), 289-358.Google Scholar
  6. 6.
    Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B.: State property systems and closure spaces: a study of categorical equivalence, International Journal of Theoretical Physics 38 (1999), 359-385.Google Scholar
  7. 7.
    Aumann, G.: Kontaktrelationen, Sitz. Ber. Bayer. Ak. Wiss., Math.-Nat. Kl. (1970), 67-77.Google Scholar
  8. 8.
    Bennett, M. K.: Affine and Projective Geometry, John Wiley and Sons, Inc., New York, 1995.Google Scholar
  9. 9.
    Birkhoff, G.: Lattice Theory, American Mathematical Society, Providence, Rhode Island, 1967.Google Scholar
  10. 10.
    Dikranjan, D., Giuli, E. and Tozzi, A.: Topological categories and closure operators. Quaestiones Mathematicae 11 (1988), 323-337.Google Scholar
  11. 11.
    Erné, M.: Lattice representations for categories of closure spaces, Categorical Topology, Sigma Series in Pure Mathematics 5, Heldermann Verlag, Berlin, 1984, pp. 197-222.Google Scholar
  12. 12.
    Faure, Cl. A. and Frölicher, A.: Morphisms of projective geometries and of corresponding lattices, Geometriae Dedicata 47 (1993), 25-40.Google Scholar
  13. 13.
    Faure, Cl. A. and Frölicher, A.: Modern Projective Geometry, Kluwer Academic Publishers, 2000.Google Scholar
  14. 14.
    Faure, Cl. A.: Categories of closure spaces and corresponding lattices, Cahier de topologie et géometrie différentielle catégoriques 35 (1994), 309-319.Google Scholar
  15. 15.
    Ganter, B. and Wille, R.: Formal Concept Analysis, Springer, Berlin, 1998.Google Scholar
  16. 16.
    Moore, D. J.: Categories of representations of physical systems, Helvetica Physica Acta 68 (1995), 658-678.Google Scholar
  17. 17.
    Moore, D. J.: Closure categories, International Journal of Theoretical Physics 36 (1997), 2707-2723.Google Scholar
  18. 18.
    Moore, D. J.: On state spaces and property lattices, Studies in the History and Philosophy of Modern Physics 30 (1999), 61-83.Google Scholar
  19. 19.
    Piron, C.: Mécanique quantique. Bases et applications, Presses polytechniques et universitaires romandes, Lausanne. Second edition (1998).Google Scholar
  20. 20.
    Preuss, G.: Theory of Topological Structures, D. Reidel Publishing Company, 1988.Google Scholar
  21. 21.
    Van der Voorde, A.: A categorical approach to T1 separation and the product of state property systems, International Journal of Theoretical Physics 39 (2000), 947-953.Google Scholar
  22. 22.
    Van der Voorde, A.: Doctoral Thesis, Brussels Free University, in preparation.Google Scholar
  23. 23.
    Van Steirteghem, B.: T0 separation in axiomatic quantum mechanics, International Journal of Theoretical Physics 39 (2000), 955-962.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • D. Aerts
    • 1
  • E. Colebunders
    • 1
  • A. Van der Voorde
    • 1
  • B. Van Steirteghem
    • 1
  1. 1.Department of MathematicsBrussels Free UniversityBrusselsBelgium

Personalised recommendations