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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
##### Keywords:
effect algebra; MacNeille completion; valuation; MV-algebra