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Nonatomic states. (English) Zbl 0960.06007
In the paper, the following definition of a nonatomic state on an orthomodular poset (OMP) is introduced: a state \(s\) on an OMP \(P\) is nonatomic if for any \(x\in P\) satisfying \(s(x)>0\) there exists \(y\in P\) such that \(y<x\) and \(0<s(y)<s(x)\). This definition coincides with the definition of a nonatomic state on a Boolean algebra. It is shown that the characterization of Boolean algebras which admit nonatomic states does not hold in the case of concrete orthomodular posets. A condition is found under which this characterization holds also for concrete orthomodular posets.
06C15 Complemented lattices, orthocomplemented lattices and posets
28A10 Real- or complex-valued set functions
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03G12 Quantum logic
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