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DOI: 10.23671/VNC.2019.21.44624 Unbounded Order Convergence and the Gordon Theorem
Emelyanov, E. Y. , Gorokhova, S. G. , Kutateladze, S. S.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 4.
Abstract:
The celebrated Gordon's theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordon's theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao-Grobler-Troitsky-Xanthos theorem saying that a sequence \(x_n\) in an Archimedean vector lattice \(X\) is \(uo\)-null (\(uo\)-Cauchy) in X if and only if \(x_n\) is \(o\)-null (\(o\)-convergent) in \(X^u\). We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially \(uo\)-complete if and only if it is \(\sigma\)-universally complete. Furthermore, we provide a comprehensive solution to Azouzi's problem on characterization of an Archimedean vector lattice in which every \(uo\)-Cauchy net is \(o\)-convergent in its universal completion.
Keywords: unbounded order convergence, universally complete vector lattice, Boolean valued analysis.
Language: English
For citation: Emelyanov, E. Y., Gorokhova, S. G., Kutateladze, S. S. Unbounded Order
Convergence and the Gordon Theorem, Vladikavkaz Math. J.,
2019, vol. 21, no. 4, pp. 56-62. DOI 10.23671/VNC.2019.21.44624
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue |
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