Quantum logics with lattice state spaces

Abstract

Let L L be a quantum logic and let S ( L ) S(L) denote the set of all states on L L . (By a state we mean a nonnegative bounded σ \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].