Truncated vector lattices: A Maeda-Ogasawara type representation

Abstract

Let L be a truncated Archimedean vector lattice whose truncation is denoted by ∗. In a recent paper, we proved that there exists a locally  compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C ∞ (X) whose truncation is  provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback,  namely, C ∞ (X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this  shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an  order dense vector sublattice of C ∞ (X) whose truncation is provided, again, by meet with some characteristic function on X.