English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

This is the second of three articles on the topic of truncation as anoperation on divisible abelian lattice-ordered groups, or simply $\ell$-groups.This article uses the notation and terminology of the first article and assumesits results. In particular, we refer to an $\ell$-group with truncation as atruncated $\ell$-group, or simply a trunc, and denote the category of truncswith truncation morphisms by $\mathbf{AT}$.  Here we develop the analog for $\mathbf{AT}$ of Madden's pointfreerepresentation for $\mathbf{W}$, the category of archimedean $\ell$-groups withdesignated order unit. More explicitly, for every archimedean trunc $A$ thereis 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 ofpointed frame maps $\mathcal{O}_{0}\mathbb{R}\rightarrow L$, and a truncisomorphism $A\rightarrow\widehat{A}$. A pointed frame map is just a frame mapbetween 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 theinsertion $0 \to \mathbb{R}$. $\left( L,\ast\right) $ is unique up to pointedframe isomorphism with respect to its properties. Finally, we reprove animportant result from the first article, namely that $\mathbf{W}$ is a non-fullmonoreflective subcategory of $\mathbf{AT}$.

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