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Quantum logics with lattice state spaces

Authors: Jiří Binder and Mirko Navara
Journal: Proc. Amer. Math. Soc. 100 (1987), 688-693
MSC: Primary 81B10; Secondary 03G12, 06C15, 46L60
MathSciNet review: 894439
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Abstract: Let $ L$ be a quantum logic and let $ S(L)$ denote the set of all states on $ L$. (By a state we mean a nonnegative bounded $ \sigma $-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

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Article copyright: © Copyright 1987 American Mathematical Society

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