Unbounded order convergence and the Gordon theorem

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.