In the category \(\textbf{V}\) of unital archimedean vector lattices, several relatively recent results have brought closure to the notion of uniform completion: there are exactly four constructs worthy of the name uniform completion. In all cases completeness requires the convergence of uniformly Cauchy sequences; the completions are distinguished by the manner in which the convergence is regulated. Ordinary uniform convergence is regulated by the canonical unit 1.
Inner relative uniform convergence, here termed iru-convergence, is regulated by an arbitrary positive element.
Outer relative uniform convergence, here termed oru-convergence, is regulated by an arbitrary positive element of a vector lattice containing the given object as a sub-vector lattice.
\(*\) -convergence is equivalent to ordinary uniform convergence on certain specified quotients of the vector lattice.
In each case the complete objects form a full monoreflective subcategory of \(\textbf{V}\) , denoted respectively \(\textbf{ucV}\) , \(\textbf{irucV}\) , \(\textbf{orucV}\) , and \(\mathbf {*cV}\) . In this article we survey these completions, comparing and contrasting them by means of a novel pointfree variant of the classical Yosida adjunction.
