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Algebraic approach to fuzzy quantum spaces. (English) Zbl 0830.03032

A new class of distributive \(\sigma\)-lattices, \(d^3\)-lattices, are introduced: a \(d^3\)-lattice is a \(\sigma\)-complete Kleene algebra where \(b\wedge \bigvee _{i=1}^\infty a_i= \bigvee_{i=1}^\infty (b\wedge a_i)\) holds. This model generalizes Boolean algebras and fuzzy quantum spaces. There is a natural congruence \(\sim\) on a \(d^3\)-lattice \(L\) such that \(L/_\sim\) is a Boolean \(\sigma\)-algebra. As a corollary, it is again shown that there is a fuzzy quantum space which admits no \(\sigma\)-additive state [see also the author and P. Pták, Fuzzy Sets Syst. 56, 123-126 (1993; Zbl 0816.28011)]. This is a negative answer to a problem posed by the reviewer [Fuzzy Sets Syst. 43, 173-181 (1991; Zbl 0742.28009)].

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
28E10 Fuzzy measure theory
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