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Proper effect algebras admitting no states. (English) Zbl 0989.81003

Summary: We show that there is even a finite proper effect algebra admitting no states. Further, every lattice effect algebra with an ordering set of valuations is an MV effect algebra (consequently it can be organized into an MV algebra). An example of a regular effect algebra admitting no ordering set of states is given. We prove that an Archimedean atomic lattice effect algebra is an MV effect algebra iff it admits an ordering set of valuations. Finally we show that every nonmodular complete effect algebra with trivial center admits no order-continuous valuations.

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C15 Complemented lattices, orthocomplemented lattices and posets
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