Archimedean l-groups with strong unit: cozero-sets and coincidence of types of ideals

Abstract

${\mathbf{W}}^{\ast }$ is the category of the archimedean l-groups with distinguished strong order unit and unit-preserving l-group homomorphisms. For $G\in {\mathbf{W}}^{\ast }$, we have the canonical compact space $YG$, and Yosida representation $G\le C(YG)$, thus, for $g\in G$, the cozero-set coz(g) in $YG$. The ideals at issue in $G$ include the principal ideals and polars, $G(g)$ and $g^{\perp \perp }$, respectively, and the ${\mathbf{W}}^{\ast }$-kernels of ${\mathbf{W}}^{\ast }$-morphisms from $G$. The ``coincidences of types" include these properties of $G$: (M) Each $G(g)=g^{\perp \perp }$; (Y) Each $G(g)$ is a ${\mathbf{W}}^{\ast }$-kernel; (CR) Each $g^{\perp \perp }$ is a ${\mathbf{W}}^{\ast }$-kernel (iff each coz(g) is regular open). For each of these, we give numerous ``rephrasings", and examples, and note that (M) = (Y) $\cap$ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on ${\mathbf{W}}^{\ast }$ brings topology to bear, and sharpens the view.