Completeness for vector lattices

A net (x(alpha))(alpha is an element of Gamma) in a vector lattice X is unbounded order convergent (uo-convergent) to x if |x(alpha) - x| Lambda y is order convergent to 0 for each y is an element of X+, and is unbounded order Cauchy (uo-Cauchy) if the net (x(alpha) - x(beta))(Gamma x Gamma) is unbounded order convergent to 0. The vector space X is unbounded order complete (uo-complete) if every uo-Cauchy net in X is uo-convergent. Recently, Li and Chen proved that a vector lattice having the countable sup property is universally complete if it is uo-complete. The main result of this paper states that this equivalence is still true without any further assumption on the vector lattice X, which answers an open question asked by Li and Chen. Some applications to this are given too. (C) 2018 Elsevier Inc. All rights reserved.