  The truncation operation facilitates the articulation and analysis of several
aspects of the structure of archimedean vector lattices; we investigate two
such aspects in this article. We refer to archimedean vector lattices equipped
with a truncation as \emph{truncs}. In the first part of the article we review
the basic definitions, state the (pointed) Yosida Representation Theorem for
truncs, and then prove a representation theorem which subsumes and extends the
(pointfree) Madden Representation Theorem. The proof has the virtue of being
much shorter than the one in the literature, but the real novelty of the
theorem lies in the fact that the topological data dual to a given trunc $G$ is
a (localic) compactification, i.e., a dense pointed frame surjection $q \colon
M \to L$ out of a compact regular pointed frame $M$. The representation is an
amalgam of the Yosida and Madden representations; the compact frame $M$ is
sufficient to describe the behavior of the bounded part $G^*$ of $G$ in the
sense that $\widetilde{G}^*$ separates the points of the compact Hausdorff
pointed space $X$ dual to $M$, while the frame $L$ is just sufficient to
capture the behavior of the unbounded part of $G$ in $\mathcal{R}_0 L$. The
truncation operation lends itself to identifying those elements of a trunc
which behave like characteristic functions, and in the second part of the
article we characterize in several ways those truncs composed of linear
combinations of such elements. Along the way, we show that the category of such
truncs is equivalent to the category of pointed Boolean spaces, and to the
category of generalized Boolean algebras. The short third part contains a
characterization of the kernels of truncation homomorphisms in terms of
pointwise closure. In it we correct an error in the literature.
