Truncated Abelian Lattice-Ordered Groups II: the Pointfree (Madden) Representation

TL;DR

This paper develops a pointfree representation for truncated abelian lattice-ordered groups, extending Madden's work, and establishes a categorical relationship between archimedean truncs and pointed frames.

Contribution

It introduces a pointfree representation for the category of truncated $ ext{l}$-groups, generalizing Madden's representation, and proves a key categorical embedding result.

Findings

Constructs a regular Lindelöf frame with a designated point for each archimedean trunc.

Establishes a truncation isomorphism between an archimedean trunc and a subtrunc of pointed frame maps.

Shows that the category of archimedean $ ext{l}$-groups with truncation is a non-full monoreflective subcategory of the category of truncs.

Abstract

This is the second of three articles on the topic of truncation as an operation on divisible abelian lattice-ordered groups, or simply $\ell$-groups. This article uses the notation and terminology of the first article and assumes its results. In particular, we refer to an $\ell$-group with truncation as a truncated $\ell$-group, or simply a trunc, and denote the category of truncs with truncation morphisms by $\mathbf{AT}$. Here we develop the analog for $\mathbf{AT}$ of Madden's pointfree representation for $\mathbf{W}$, the category of archimedean $\ell$-groups with designated order unit. More explicitly, for every archimedean trunc $A$ there is a regular Lindel\"{o}f frame $L$ equipped with a designated point $\ast : L \rightarrow 2$, a subtrunc $\widehat{A}$ of $\mathcal{R}_{0}L$, the trunc of pointed frame maps $\mathcal{O}_{0}\mathbb{R}\rightarrow L$, and a trunc isomorphism $A\rightarrow\widehat{A}$. A pointed frame map is just a frame map between frames which commutes with their designated points, and $\mathcal{O}_{0}\mathbb{R}$ stands for the pointed frame which is the topology $\mathcal{O}\mathbb{R}$ of the real numbers equipped with the frame map of the insertion $0 \to \mathbb{R}$. $\left( L,\ast\right) $ is unique up to pointed frame isomorphism with respect to its properties. Finally, we reprove an important result from the first article, namely that $\mathbf{W}$ is a non-full monoreflective subcategory of $\mathbf{AT}$.