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The truncation operation facilitates the articulation and analysis of severalaspects of the structure of archimedean vector lattices; we investigate twosuch aspects in this article. We refer to archimedean vector lattices equippedwith a truncation as \emph{truncs}. In the first part of the article we reviewthe basic definitions, state the (pointed) Yosida Representation Theorem fortruncs, and then prove a representation theorem which subsumes and extends the(pointfree) Madden Representation Theorem. The proof has the virtue of beingmuch shorter than the one in the literature, but the real novelty of thetheorem lies in the fact that the topological data dual to a given trunc $G$ isa (localic) compactification, i.e., a dense pointed frame surjection $q \colonM \to L$ out of a compact regular pointed frame $M$. The representation is anamalgam of the Yosida and Madden representations; the compact frame $M$ issufficient to describe the behavior of the bounded part $G^*$ of $G$ in thesense that $\widetilde{G}^*$ separates the points of the compact Hausdorffpointed space $X$ dual to $M$, while the frame $L$ is just sufficient tocapture the behavior of the unbounded part of $G$ in $\mathcal{R}_0 L$. Thetruncation operation lends itself to identifying those elements of a truncwhich behave like characteristic functions, and in the second part of thearticle we characterize in several ways those truncs composed of linearcombinations of such elements. Along the way, we show that the category of suchtruncs is equivalent to the category of pointed Boolean spaces, and to thecategory of generalized Boolean algebras. The short third part contains acharacterization of the kernels of truncation homomorphisms in terms ofpointwise closure. In it we correct an error in the literature.

Structural aspects of truncated archimedean vector lattices: simple elements, good sequences