Within archimedean l-groups, “G ∈ SW∗” means there are H with strong unit (H ∈ W∗) and an embedding G ≤ H. A. Theorem (6.1). For X a Tychonoff space (or completely regular locale), C(X) ∈ SW∗ iff X is pseudocompact (which means C(X) ∈ W∗). But any G ∈ SW∗ embeds into various C(X), and any C(X) contains many H ∈ W∗. We define cardinal invariants $\mathfrak {b}G$ , $\mathfrak {d}G$ , λG which generalize respectively, the bounding and dominating numbers for $\mathbb {R}^{\mathbb {N}}$ , $\mathfrak {b}$ and $\mathfrak {d}$ , and the π-weight of a topological space. B. Theorem (6.3, 7.5). SupposeG ≤ C(X), and X contains densely $\bigcup _{I} X_{i}$ , the Xi compact. Then G ∈ SW∗ if either (|I| = ω and $\mathfrak {d}G < \mathfrak {b}$ ) or ( $|I| < \mathfrak {b}$ and $\mathfrak {d}G = \omega $ ). This “ $\omega , \mathfrak {b}$ symmetry” fades a bit in the following, where $\mathfrak {p}$ is the pseudo-intersection number for $\mathbb {R}^{\mathbb {N}}$ ( $\mathfrak {p} \le \mathfrak {b}$ , with = under Martin’s Axiom). C. Theorem (8.2). G ∈ SW∗ if either $(\mathfrak {d}G = \omega $ and $\lambda G < \mathfrak {b}$ ) or ( $\mathfrak {d}G < \mathfrak {p}$ andλG = ω). Examples (§9) show limits on the hypotheses in B and C.
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