Some minimum topological spaces, and vector lattices

TL;DR

The work analyzes when a minimum representation exists among covers of a fixed compact space $K$ under a covering operator $c$, and translates this to hull-type completions in Archimedean vector lattices via the Yosida duality. By constructing the Yosida bridge $\mu(c)$ and identifying fundamental instances $\mu(g)=e$ and $\mu(\mathrm{id})=u$, the authors obtain explicit minima for several operators (identity, Gleason $g$, atom $a(\gamma)$, and quasi-$F$ $qF$) and characterize when minima exist in the associated hull frameworks, including zero-dimensional cases and $F(K)$-type minima. They also derive a parallel result for Boolean algebras under completion, showing minima occur precisely when the base algebra is a power set, with the minimum given by the finite/cofinite subalgebra. The paper further establishes a transfer principle between the covering and hull settings via the Yosida space, clarifying how minima in $\mathcal{S}(K,c)$ correspond to minima in $\mathcal{V}(H,h)$ and providing concrete instances of this correspondence.

Abstract

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).

Theorems & Definitions (53)

• Theorem 2.1
• Definition 2.2
• Theorem 3.1
• Definition 3.2
• Theorem 4.1
• Remark 4.2
• Definition 4.3
• Theorem 4.4
• proof
• Theorem 4.5