Navara, Mirko Algebraic approach to fuzzy quantum spaces. (English) Zbl 0830.03032 Demonstr. Math. 27, No. 3-4, 589-600 (1994). 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)]. Reviewer: A.Dvurečenskij (Bratislava) Cited in 2 Documents MSC: 03G12 Quantum logic 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 28E10 Fuzzy measure theory Keywords:distributive \(\sigma\)-lattices; \(d^ 3\)-lattices; \(\sigma\)-complete Kleene algebra; Boolean algebras; fuzzy quantum spaces Citations:Zbl 0816.28011; Zbl 0742.28009 PDFBibTeX XMLCite \textit{M. Navara}, Demonstr. Math. 27, No. 3--4, 589--600 (1994; Zbl 0830.03032) Full Text: DOI