International Journal of Theoretical Physics

, Volume 31, Issue 5, pp 881–888

Modular almost orthogonal quantum logics

  • Sylvia Pulmannová
  • Zdenka Riečanová
Article

Abstract

Almost orthogonal quantum logics, i.e., atomic orthomodular lattices in which to every atom there exist only finitely many nonorthogonal atoms, are studied. It is shown that an almost orthogonal quantum logic is modular if and only if it has the exchange property if and only if it can be embedded into a direct product of finite modular quantum logics. The class of almost orthogonal modular OMLs is the largest subclass of the class of atomic modular OMLs in which the conditions commutator-finite and block-finite are equivalent. A finite faithful valuation on an almost orthogonal quantum logicL exists if and only ifL is modular and the set of all atoms ofL is at most countable.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Birkhoff, G. (1967).Lattice Theory, American Mathematical Society, Providence, Rhode Island.Google Scholar
  2. Erné, M. and Weck, S. (1980). Order convergence in lattices,Rocky Mountain Journal of Mathematics,10, 805–818.Google Scholar
  3. Greechie, R., and Herman, L. (1985). Commutator-finite orthomodular lattices,Order,1, 277–284.Google Scholar
  4. Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York.Google Scholar
  5. Maeda, F., and Maeda, S. (1970).Theory of Symmetric Lattices, Springer-Verlag, Berlin.Google Scholar
  6. Pulmannová, S., and Riečanová, Z. (1990a). A remark to orthomodular lattices, inProceedings of the 2nd Winter School on Measure Theory, Liptovský Ján, pp. 175–176.Google Scholar
  7. Pulmannová, S., and Riečanová, Z. (1990b). Compact topological orthomodular lattices, inContributions to General Algebra 7, Proceedings of the Vienna Conference, June 14–17, 1990, pp. 277–282.Google Scholar
  8. Pulmannová, S., and Rogalewicz, V. (1991). Orthomodular lattices with almost orthogonal sets of atoms,Commentationes Mathematicae Universitatis Carolinae,32, pp. 423–429.Google Scholar
  9. Riečanová, Z. (1989). Topologies in atomic quantum logics,Acta Universitatis Carolinae Mathematica et Physica,30, 143–148.Google Scholar
  10. Riečanová, Z. (1990). On the MacNeille completion of (o)-continuous atomic logics, inProceedings of the 2nd Winter School on Measure Theory, Liptovský Ján, pp. 182–187.Google Scholar
  11. Riečanová, Z. (1991). Application of topological methods to the completion of atomic orthomodular lattices,Demonstratio Mathematica,XXIV (1–2), pp. 331–341.Google Scholar
  12. Sarymsakov, T. A., Ajupov, S. A., Chadžijev, D., and Chilin, V. J. (1983).Order Algebras, FAN, Tashkent [in Russian].Google Scholar
  13. Tae Ho Choe, and Greechie, R. (to appear). Profinite orthomodular lattices,Proceedings of the American Mathematical Society.Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Sylvia Pulmannová
    • 1
  • Zdenka Riečanová
    • 2
  1. 1.Mathematical InstituteSlovak Academy of ScienceBratislavaCzechoslovakia
  2. 2.Department of Mathematics, Electrotechnical FacultySlovak Technical UniversityBratislavaCzechoslovakia

Personalised recommendations