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.