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Archimedean atomic lattice effect algebras with complete lattice of sharp elements. (English) Zbl 1189.03073

Summary: We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such an effect algebra \(E\) is separable and modular then there exists a faithful state on \(E\). Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra \(E\) and the compatiblity center of \(E\) is not a Boolean algebra then there exists an \((o)\)-continuous subadditive state on \(E\).

MSC:

03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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