Adjoining a Strong Unit to an Archimedean Lattice-Ordered Group

Abstract

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 X_i 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.