Completeness for vector lattices

Abstract

A net ( x α ) α ∈ Γ in a vector lattice X is unbounded order convergent (uo-convergent) to x if | x α − x | ∧ y is order convergent to 0 for each y ∈ X + , and is unbounded order Cauchy (uo-Cauchy) if the net ( x α − x β ) Γ × Γ 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.