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Weak relatively uniform convergence in Riesz groups. (English) Zbl 1236.06021
Chajda, I. (ed.) et al., Proceedings of the 79th workshop on general algebra “79. Arbeitstagung Allgemeine Algebra”, 25th conference of young algebraists, Palacký University Olomouc, Olomouc, Czech Republic, February 12–14, 2010. Klagenfurt: Verlag Johannes Heyn (ISBN 978-3-7084-0407-3/pbk). Contributions to General Algebra 19, 127-138 (2010).
Weak relatively uniform convergences in abelian lattice-ordered groups were introduced and investigated by Š. Černák and J. Jakubík [Math. Slovaca 61, No. 5, 687–704 (2011; Zbl 1274.06061)]. The author of the paper under review remarks that the the definitions of Černák and Jakubík given in the mentioned paper can be used with a small adaptation also for Riesz groups. He shows that several basic properties of weak relatively uniform convergences in abelian lattice-ordered groups remain valid for the case of Riesz groups. E.g., let us consider sequences $$(x_n)$$ and $$(y_n)$$ in an abelian lattice-ordered group $$G$$ such that $$x_n\leq y_n$$ for every $$n$$. Let $$\alpha$$ be a fixed weak relatively uniform convergence in $$G$$ such that $$x_n@>\alpha>> x$$, $$y_n@>\alpha>> y$$. Then $$x\leq y$$. In the paper under review it is shown that an analogous result fails to be valid in general for abelian Archimedean Riesz groups. Further, a Cauchy completion of an isolated abelian Riesz group with a weak relatively uniform convergence is described.
For the entire collection see [Zbl 1201.08001].
MSC:
 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces