The four uniform completions of a unital archimedean vector lattice

TL;DR

This paper explores four different notions of uniform completeness in unital archimedean vector lattices, characterizing their completions through a unified pointfree approach and establishing their categorical properties.

Contribution

It introduces a unified framework for understanding four types of uniform completions using a novel pointfree Yosida adjunction in the category of vector lattices.

Findings

Four notions of uniform completeness are characterized.

Each completion forms a full monoreflective subcategory.

A unified pointfree approach to these completions is developed.

Abstract

In the category \(\mathbf{V}\) of unital archimedean vector lattices, four notions of uniform completeness obtain. 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 \(\mathbf{V}\), denoted respectively \(\mathbf{ucV}\), \(\mathbf{irucV}\), \(\mathbf{orucV}\), and \(\mathbf{*cV}\). In this article we provide a unified development of these completions by means of a novel pointfree variant of the classical Yosida adjunction.