  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 $\mathbf{AT}$.
