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Distributive atomic effect algebras. (English) Zbl 1039.03050
It is proved that the MacNeille completion of an Archimedean atomic distributive effect algebra is a direct product of finite chains and distributive diamonds. As a consequence, for such effect algebras \(E\) it is proved: (1) every faithful or \((o)\)-continuous state on \(E\) is a valuation (hence a subadditive state); (2) there is an \((o)\)-continuous valuation on \(E\), (3) if \(E\) is complete then it is a homomorphic image of a complete atomic modular ortholattice.

MSC:
03G12 Quantum logic
06D35 MV-algebras
06C15 Complemented lattices, orthocomplemented lattices and posets
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