Truncated abelian lattice-ordered groups I: The pointed (Yosida) representation

Abstract

This article is about truncation as an operation on divisible abelian lattice-ordered groups (ℓ-groups). Suppose A is an ℓ-subgroup of an ℓ-group B, and suppose 0⩽u∈B is such that no element of A is either disjoint from u, or infinitesimal with respect to u. If a∧u∈A for all a∈A+ then we say that A is closed under truncation by u. We do not assume that u is present in A, nor do we assume any knowledge of B. In effect, truncation by u constitutes a unary operation on A.We present several axiom systems for truncation and show them equivalent, thus defining the category T of ℓ-groups with truncation, which we call truncated ℓ-groups, or truncs for short. The morphisms of T are the ℓ-homomorphisms which preserve truncation. A trunc A is called unital if it happens to contain an element u⩾0 such that the truncation is provided by meet with u.We show that every trunc A is associated with a compact Hausdorff space X having a designated point ⁎. If A is archimedean then there is a trunc Aˆ which separates points from closed sets in D⁎X, the family of almost-finite extended-real valued functions on X which vanish at ⁎, and there is a trunc isomorphism A→Aˆ. The space X is unique up to pointed homeomorphism with respect to its properties. This is the direct generalization to archimedean truncs of the classical Yosida representation for W, which appears here as the full subcategory of T consisting of the unital archimedean truncs. We show W to be bireflective in T.We conclude by considering the important question of which divisible archimedean ℓ-groups support a truncation. We point out an example in the literature of one which does not, and we note a couple of fairly extensive classes of divisible archimedean ℓ-groups which do support a truncation. Finally, we observe that any trunc has what is known as a Johnson representation.