Quantum logics with lattice state spaces

Binder, Jiří; Navara, Mirko

doi:10.1090/S0002-9939-1987-0894439-1

Quantum logics with lattice state spaces
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by Jiří Binder and Mirko Navara PDF

Proc. Amer. Math. Soc. 100 (1987), 688-693 Request permission

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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