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, $\mathbf{W}$), where the Yosida representation $A\le D(Y(A,u))$ is available. Here, $A$ SMP is equivalent to $Y(A,u)$ having a $\ddot{I}€$-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 $\ddot{I}€$-base, a property we call $\ddot{I}€$-zero-dimensional ($\ddot{I}€$ZD). The paper is in two parts: the first explicates the similarities of SMP and $\ddot{I}€$ZD; the second consists of examples, including $\ddot{I}€$ZD but not SMP, and constructions of many SMP's which seem scarce in the literature.