A. W. Hager | Searcharxiv

arXiv2026

Some minimum topological spaces, and vector lattices

We investigate the existence of compact Hausdorff spaces $X$ that are minimum with respect to $cX=K$ for some fixed covering operator $c$ and compact Hausdorff space $K$ with $cK=K$. Then, using the Yosida representation theorem, we show how that situation relates to the existence of Archimedean vector lattices $A$ with distinguished strong unit that are minimum with respect to $hA=H$ for some fixed hull operator $h$ and vector lattice $H$ with $hH=H$. Among others, we obtain answers for $c=g$ (the Gleason covering operator), $c=qF$ (the quasi-$F$ covering operator), $h = u$ (the uniform completion operator), and $h=e$ (the essential completion operator).

math.FA↗

arXiv2024

The four uniform completions of a unital archimedean vector lattice

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.

math.FA↗