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

Dvurečenskij, Anatolij; Vetterlein, Thomas

Algebra Univers. 50, No. 2, 207-230 (2003).

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.

Reviewer: Jan Kühr (Olomouc)

MSC:

03G12	Quantum logic
06F15	Ordered groups

Keywords:

pseudoeffect algebras; po-groups; PE-algebras with Riesz properties; Archimedean PE-algebras; sup-homogeneous PE-algebras; MacNeille completion of PE-algebras