Sufficiently many projections in archimedean vector lattices with weak order unit

TL;DR

This paper explores the conditions under which archimedean vector lattices with a weak order unit have sufficiently many projections, linking algebraic properties to topological bases, and provides examples and constructions related to these properties.

Contribution

It establishes the equivalence between the SMP property and the existence of a certain type of clopen base in the Yosida representation, and constructs new examples of such lattices.

Findings

SMP is equivalent to Y(A,u) having a π-base of local clopen sets.

Strong units imply SMP if and only if Y(A,u) has a π-zero-dimensional base.

Constructs numerous examples of SMPs and distinguishes them from πZD.

Abstract

The property of a vector lattice of sufficiently many projections (SMP) is informed by restricting attention to archimedean $A$ with a distinguished weak order unit $u$ (the class, or category, $\bf{W}$), where the Yosida representation $A \leq D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to $Y(A,u)$ having a $\pi$-base of clopen sets of a certain type called ``local". If the unit is strong, all clopen sets are local and $A$ is SMP if and only if $Y(A,u)$ has clopen $\pi$-base, a property we call $\pi$-zero-dimensional ($\pi$ZD). The paper is in two parts: the first explicates the similarities of SMP and $\pi$ZD; the second consists of examples, including $\pi$ZD but not SMP, and constructions of many SMP's which seem scarce in the literature.