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 be a quantum logic and let denote the set of all states on . (By a state we mean a nonnegative bounded -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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DOI:
https://doi.org/10.1090/S0002-9939-1987-0894439-1

Article copyright:
© Copyright 1987
American Mathematical Society