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On the concreteness of quantum logics. (English) Zbl 0586.03050
The authors take the very conventional view of quantum logic as an orthomodular poset and conduct a very traditional enquiry into its relation to classical Boolean logic. That is, they attempt to find a natural relation between the structure of this orthomodular poset and the classical structure of a subset of sets. The meagreness of their results should serve to show that this traditional approach has proved unfruitful. The authors do not attempt to relate their discussion to quantum mechanics.
In fact the authors establish a relation between the ”centres”, i.e. the globally compatible elements, of the orthomodular posets and the centres of ”concrete” logics, i.e. of orthocomplemented posets with 1 and 0 which are isomorphic to a collection of subsets of a set. This result will probably be of interest only to the most devoted ”quantum logicians”. They also, perhaps more interestingly, establish an embedding of orthocomplemented posets into concrete logics.
Reviewer: R.Wallace Garden

03G12 Quantum logic
Full Text: EuDML
[1] V. Alda: On 0-1 measures for projectors. Aplikace Matematiky 26, 57-58 (1981). · Zbl 0459.28020
[2] L. J. Bunce D. M. Wright: Qantum measures and states on Jordan algebras. Comm. Math. Phys. · Zbl 0579.46049
[3] J. Brabec P. Pták: On compatibility in quantum logics. Foundations of Physics, Vol. 12, No. 2, 207-212 (1982).
[4] R. Godowski: Varieties of orthomodular lattices with a strongly full set of states. Demonstration Mathematica, Vol. XIV, No. 3) · Zbl 0483.06007
[5] R. Greechie: Orthomodular lattices admitting no states. J. Comb. Theory 10, 119-132 (1971). · Zbl 0219.06007
[6] S. Gudder: Stochastic Methods in Quantum Mechanics. North-Holland 1979. · Zbl 0439.46047
[7] P. Pták: Weak dispersion-free states and the hidden variables hypothesis. J. Math. Physics 24 (4), 839-840(1983). · Zbl 0508.60006
[8] P. Pták V. Rogolewicz: Measures on orthomodular partially ordered sets. J. Pure and Applied Algebra 28, 75-85 (1983). · Zbl 0507.06008
[9] S. Pulmannová: Compatibility and partial compatibility in quantum logics. Ann. Inst. Henri Poincaré, Vol. XXXIV, No. 4, 391-403 (1981). · Zbl 0469.03045
[10] R. Sikorski: Boolean Algebras. Springer-Verlag (1964). · Zbl 0123.01303
[11] V. Varadarajan: Geometry of Quantum Theory I. Von Nostrand, Princeton (1968). · Zbl 0155.56802
[12] M. Zierler M. Schlessinger: Boolean embedding of orthomodular sets and quantum logics. Duke J. Math. 32, 251-262 (1965). · Zbl 0171.25403
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