  $\bf{W}^*$ is the category of the archimedean l-groups with distinguished
strong order unit and unit-preserving l-group homomorphisms. For $G \in
\bf{W}^*$, we have the canonical compact space $YG$, and Yosida representation
$G \leq C(YG)$, 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 $\bf{W}^*$-kernels of $\bf{W}^*$-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 $\bf{W}^*$-kernel; (CR) Each
$g^{\perp \perp}$ is a $\bf{W}^*$-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 $\bf{W}^*$ brings topology to bear, and
sharpens the view.
