The Freudenthal spectral theorem and sufficiently many projections in Archimedean vector lattices

The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each \(0 < e \in A\) $0<e\in A$ , the Yosida space for \((e^{dd},e)\) $(e^{\mathit{dd}},e)$ , denoted \(Y_e\) $Y_e$ , has a base of clopen sets. This yields a short “Yosida based" proof of FST. On the other hand, SMP implies that each \(Y_e\) $Y_e$ has a \(\pi \) $\pi$ -base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation).