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.
Similar content being viewed by others
References
Birkhoff, G. (1967).Lattice Theory, American Mathematical Society, Providence, Rhode Island.
Erné, M. and Weck, S. (1980). Order convergence in lattices,Rocky Mountain Journal of Mathematics,10, 805–818.
Greechie, R., and Herman, L. (1985). Commutator-finite orthomodular lattices,Order,1, 277–284.
Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York.
Maeda, F., and Maeda, S. (1970).Theory of Symmetric Lattices, Springer-Verlag, Berlin.
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.
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.
Pulmannová, S., and Rogalewicz, V. (1991). Orthomodular lattices with almost orthogonal sets of atoms,Commentationes Mathematicae Universitatis Carolinae,32, pp. 423–429.
Riečanová, Z. (1989). Topologies in atomic quantum logics,Acta Universitatis Carolinae Mathematica et Physica,30, 143–148.
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.
Riečanová, Z. (1991). Application of topological methods to the completion of atomic orthomodular lattices,Demonstratio Mathematica,XXIV (1–2), pp. 331–341.
Sarymsakov, T. A., Ajupov, S. A., Chadžijev, D., and Chilin, V. J. (1983).Order Algebras, FAN, Tashkent [in Russian].
Tae Ho Choe, and Greechie, R. (to appear). Profinite orthomodular lattices,Proceedings of the American Mathematical Society.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pulmannová, S., Riečanová, Z. Modular almost orthogonal quantum logics. Int J Theor Phys 31, 881–888 (1992). https://doi.org/10.1007/BF00678552
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00678552