In the paper, we revisit several approaches to the concept of uniform completion \(X^{\textrm{ru}}\) of a vector lattice X. We show that many of these approaches yield the same result. In particular, if X is a sublattice of a uniformly complete vector lattice Z then \(X^{\textrm{ru}}\) may be viewed as the intersection of all uniformly complete sublattices of Z containing X. \(X^{\textrm{ru}}\) may also be constructed via a transfinite process of taking uniform adherences in Z with regulators coming from the previous adherences. If, in addition, X is majorizing in Z then \(X^{\textrm{ru}}\) may be viewed as the uniform closure of X in Z. We show that \(X^{\textrm{ru}}\) may also be characterized via a universal property: every positive operator from X to a uniformly complete vector lattice extends uniquely to \(X^{\textrm{ru}}\) . Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.
