Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Some nearly Boolean orthomodular posets


Author: Pavel Pták
Journal: Proc. Amer. Math. Soc. 126 (1998), 2039-2046
MSC (1991): Primary 28A60, 06C15, 81P10
DOI: https://doi.org/10.1090/S0002-9939-98-04403-7
MathSciNet review: 1452822
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $L$ be an orthomodular partially ordered set (``a quantum logic"). Let us say that $L$ is nearly Boolean if $L$ is set-representable and if every state on $L$ is subadditive. We first discuss conditions under which a nearly Boolean OMP must be Boolean. Then we show that in general a nearly Boolean OMP does not have to be Boolean. Moreover, we prove that an arbitrary Boolean algebra may serve as the centre of a (non-Boolean) nearly Boolean OMP.


References [Enhancements On Off] (What's this?)

  • [1] Beltrametti, E., Cassinelli, G., The logic of quantum mechanics, Addison-Wesley, Reading, Massachusetts, 1981. MR 83d:81008
  • [2] Bruns, G., Greechie, R.J., Harding, J., Roddy, M., Completions of orthomodular lattices, Order 7 (1990), 67-76. MR 92b:06029
  • [3] DeLucia, P., Pták, P., Quantum probability spaces that are nearly classical, Bull. Polish Acad. Sciences - Math. 40 (2) (1992), 163-173. MR 97i:81015
  • [4] Gudder, S., Stochastic Methods in Quantum Mechanics, North-Holland, Amsterdam, 1979. MR 84j:81003
  • [5] Halmos, P., Measure Theory, Van Nostrand, New York, 1950. MR 11:504d
  • [6] Müller, V., Jauch-Piron states on concrete quantum logics, Int. Journ. Theor. Phys. 32 (3) (1993), 433-442. MR 94g:81011
  • [7] Navara, M., Kernel logics, Czechoslovak Math. J., (to appear).
  • [8] Navara, M., Pták, P., Almost Boolean orthomodular posets, Jour. Pure Applied Algebra 60 (1989), 105-111. MR 90h:06003
  • [9] Pták, P., Logics with given centers and state spaces, Proc. Amer. Math. Soc. 88 (1983), 106-109. MR 84f:06016
  • [10] Pták, P., Summing of Boolean algebras and logics, Demonstratio Math. 19 (1986), 349-357. MR 89b:03104
  • [11] Pták, P., Jauch-Piron property (everywhere!) in the logico-algebraic foundation of quantum theories, Int. Journ. Theor. Phys. 32 (10) (1993), 1985-1991. MR 94j:81024
  • [12] Müller, V., Pták, P., Tkadlec, J., Concrete quantum logics with covering properties, Int. Journ. Theor. Phys. 31 (5) (1992), 843-854. MR 93a:81021
  • [13] Pták, P., Pulmannová, S., Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht/Boston/London, 1991. MR 94d:81018b
  • [14] Pták, P., Pulmannová, S., A measure theoretic characterization of Boolean algebras among orthomodular lattices, Comment. Math. Univ. Carolinae 35 (1) (1994), 205-208. MR 95i:06014
  • [15] Sikorski, R., Boolean Algebras, Springer Verlag, Berlin-Heidelberg-New York, 1969. MR 39:4053
  • [16] Varadarajan, V., Geometry of Quantum Theory I, Van Nostrand, Princeton, 1968. MR 57:11399

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28A60, 06C15, 81P10

Retrieve articles in all journals with MSC (1991): 28A60, 06C15, 81P10


Additional Information

Pavel Pták
Affiliation: Czech Technical University, Faculty of Electrical Engineering, Department of Mathematics, 16627 Prague 6, Czech Republic
Email: ptak@math.feld.cvut.cz

DOI: https://doi.org/10.1090/S0002-9939-98-04403-7
Keywords: Orthomodular partially ordered set, Boolean algebra, state (= finitely additive probability measure), subadditivity
Received by editor(s): December 16, 1996
Additional Notes: The author acknowledges the support by the grant GA 201/96/0117 of the Czech Grant Agency.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society