$\alpha$-Projectable and laterally $\alpha$-complete Archimedean lattice-ordered groups with weak unit via topology

Abstract

Let $\bf{W}$ be the category of Archimedean lattice-ordered groups with weak order unit, $\bf{Comp}$ the category of compact Hausdorff spaces, and $\mathbf{W} \xrightarrow{Y} \mathbf{Comp}$ the Yosida functor, which represents a $\bf{W}$-object $A$ as consisting of extended real-valued functions $A \leq D(YA)$ and uniquely for various features. This yields topological mirrors for various algebraic ($\bf{W}$-theoretic) properties providing close analysis of the latter. We apply this to the subclasses of $\alpha$-projectable, and laterally $\alpha$-complete objects, denoted $P(\alpha)$ and $L(\alpha)$, where $\alpha$ is a regular infinite cardinal or $\infty$. Each $\bf{W}$-object $A$ has unique minimum essential extensions $A \leq p(\alpha) A \leq l(\alpha) A$ in the classes $P(\alpha)$ and $L(\alpha)$, respectively, and the spaces $Yp(\alpha) A$ and $Yl(\alpha) A$ are recognizable (for the most part); then we write down what $p(\alpha) A$ and $l(\alpha) A$ are as functions on these spaces. The operators $p(\alpha)$ and $l(\alpha)$ are compared: we show that both preserve closure under all implicit functorial operations which are finitary. The cases of $A = C(X)$ receive special attention. In particular, if ($\omega < \alpha$) $l(\alpha) C(X) = C(Yl(\alpha) C(X))$, then $X$ is finite. But ($\omega \leq \alpha$) for infinite $X$, $p(\alpha) C(X)$ sometimes is, and sometimes is not, $C(Yp(\alpha) C(X))$.

Keywords

• lattice-ordered group
• archimedean
• projectable
• laterally complete