## Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups.(English)Zbl 1092.03034

Pseudoeffect algebras are a common generalization of effect algebras and pseudo MV-algebras; they are equipped with a partial addition which is non-commutative in general. Typical examples of pseudoeffect algebras (PE-algebras) arise from intervals in partially ordered groups (po-groups), but not all PE-algebras are obtained in this form. The authors construct the MacNeille completion of PE-algebras and they prove that this completion of a PE-algebra $$E$$ is a PE-algebra if and only if $$E$$ fulfills the subset closedness property, which yields the archimedeanness of $$E$$. In the particular case when $$E$$ is a pseudo MV-algebra, $$E$$ possesses a PE-MacNeille completion if and of if it is archimedean. The rest of the paper is devoted to the relationships between archimedeanness of interval PE-algebras and archimedeanness of their representing po-groups. It is shown that a sup-homogeneous PE-algebra satisfying a certain version of the Riesz decomposition property is archimedean if and only if so is its representing po-group.

### MSC:

 03G12 Quantum logic 06F15 Ordered groups
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