Truncated abelian lattice-ordered groups II: the pointfree (Madden) representation

Abstract

This is the second of three articles on the topic of truncation as an operation on divisible abelian lattice-ordered groups, or simply ℓ-groups. This article uses the notation and terminology of the first article and assumes its results. In particular, we refer to an ℓ-group with truncation as a truncated ℓ-group, or simply a trunc, and denote the category of truncs with truncation homomorphisms by AT.Here we develop the analog for AT of Madden's pointfree representation for W, the category of archimedean ℓ-groups with designated order unit. More explicitly, for every archimedean trunc A there is a regular Lindelöf frame L equipped with a designated point ⁎:L→2, a subtrunc Aˆ of R0L, the trunc of pointed frame maps O0R→L, and a trunc isomorphism A→Aˆ. A pointed frame map is just a frame map between frames which commutes with their designated points, and O0R stands for the pointed frame which is the topology OR of the real numbers equipped with the frame map of the insertion 0→R. (L,⁎) is unique up to pointed frame isomorphism with respect to its properties. Finally, we reprove an important result from the first article, namely that W is a non-full monoreflective subcategory of AT.