Sufficiently many projections in archimedean vector lattices with weak unit

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