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A measure-theoretic characterization of Boolean algebras among orthomodular lattices. (English) Zbl 0805.06010

An orthomodular lattice \(L\) is called unital with respect to subadditive states if for any non-zero \(a\in L\) there exists a subadditive state \(s\) on \(L\) such that \(s(a)= 1\). (Recall that “state” is an alternative name for a probability measure and that a state is subadditive if \(s(a\vee b)\leq s(a)+ s(b)\) holds for any \(a,b\in L\).) It is shown that Boolean algebras are the only orthomodular lattices unital with respect to subadditive states.

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
28E15 Other connections with logic and set theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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