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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.
##### MSC:
 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
##### Keywords:
concrete orthomodular poset; nonatomic state; embedding
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##### References:
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