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Fuzzy quantum spaces and compatibility. (English) Zbl 0657.60004
Using ideas from mathematics for fuzzy systems, the authors generalize the notion of probability quantum space introduced by P. Suppes [see Philos. Sci. 33, 14-21 (1966)] to so-called fuzzy quantum spaces. The main goal of their paper is to present conditions for the ranges of observables in a fuzzy quantum space to have classical character, i.e. to be embeddable into some Boolean $$\sigma$$-algebra.
This question is known as the compatibility problem and it has been solved for different classes of quantum logics using different notions of compatibility. Here it is shown that for finite compatible observables in any fuzzy quantum space joint-observables always exist.
Reviewer: K.Piasecki

##### MSC:
 60A99 Foundations of probability theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 03E72 Theory of fuzzy sets, etc.
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##### References:
 [1] Brabec, J. (1979). Compatibility in orthomodular posets,Časopis pro pěstování matematiky,104, 149–153. · Zbl 0416.06004 [2] Brabec, J., and Pták, P. (1982). On compatiblity in quantum logics,Foundations of Physics,12, 207–212. [3] Dvurečenskij, A., and Riečan, B. (submitted). On joint observables forF-quantum spaces, submitted for publication. [4] Gudder, S. P. (1979).Stochastic Methods in Quantum Mechanics, Elsevier/North-Holland, Amsterdam. · Zbl 0439.46047 [5] Guz, W. (1985). Fuzzy{$$\sigma$$}-algebras of physics,International Journal of Theoretical Physics,24, 481–493. · Zbl 0575.46052 [6] Kolmogorov, A. N. (1933).Grundebegriffe der Wahrscheinlichkeitsrechnung, Berlin. · JFM 59.1154.01 [7] Neubrunn, T. (1970). A note on quantum probability spaces.Proceedings of the American Mathematical Society,25, 672–675. · Zbl 0208.43402 [8] Neubrunn, T., and Pulmannová, S. (1983). On compatibility in quantum logics,Acta Mathematica Universitatis Comenianae,42/43, 153–168. · Zbl 0539.03045 [9] Piasecki, K. (1985). Probability of fuzzy events defined as denumerable additivity measure,Fuzzy Sets and Systems,17, 271–284. · Zbl 0604.60005 [10] Pykacz, J. (1987). Quantum logics and soft fuzzy probability spaces,Busefal,32 (10). · Zbl 0662.03055 [11] Riečan, B., and Dvurečenskij, A. (to appear). Randomness and fuzziness, in:Progress in Fuzzy Sets in Europe, to appear. [12] Sikorski, R. (dy1964).Boolean Algebras, Springer, Verlag. · Zbl 0123.01303 [13] Suppes, P. (1966). The probability argument for a non-classical logic of quantum mechanics,Philosophy of Science,33, 14–21. [14] Varadarajan, V. S. (1962). Probability in physics and a theorem on simultaneous observability.Communications in Pure and Applied Mathematics,15, 189–217 [Errata,18 (1965)]. · Zbl 0109.44705 [15] von Neumann, J. (1932).Grundlagen der Quantummechanik, Springer, Berlin. · Zbl 0005.09104
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