Sufficiently many projections in archimedean vector lattices with weak order unit

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.