Unbounded order convergence and the Gordon theorem#

Creative Commons License

EMELYANOV E., Gorokhova S., Kutateladze S.

Vladikavkaz Mathematical Journal, vol.21, no.4, pp.56-62, 2019 (Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 21 Issue: 4
  • Publication Date: 2019
  • Doi Number: 10.23671/vnc.2019.21.44624
  • Journal Name: Vladikavkaz Mathematical Journal
  • Journal Indexes: Scopus
  • Page Numbers: pp.56-62
  • Keywords: Boolean valued analysis, Unbounded order convergence, Universally complete vector lattice
  • Middle East Technical University Affiliated: Yes


© 2019 Emelyanov, E. Y., Gorokhova, S. G. and Kutateladze, S. S.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 xn in an Archimedean vector lattice X is uo-null (uo-Cauchy) in X if and only if xn is o-null (o-convergent) in Xu. 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 σ-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.