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.


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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