When Max_d(G) is zero-dimensional
Ricardo Carrera, Ramiro Lafuente-Rodriguez, Warren Wm. McGovern
TL;DR
The paper addresses when the space of maximal $d$-subgroups $Max_d(G)$ of a W-object $(G,u)$ is zero-dimensional, extending prior results on the Yosida space and related bases. It develops the interplay between $Max_d(G)$, $Max_c(G)$, and the Wallman lattice $\mathscr{G}(G)$ via the surjective map $\rho: Max_d(G) \to Max_c(G)$ and the Yosida representation, establishing that $Max_c(G)$ corresponds to $\mathrm{Ult}(\mathscr{G}(G))$ and that the zero-dimensionality of $Max_d(G)$ is characterized by several equivalent conditions including separation by complemented elements, injectivity of $\rho$, and base properties on $YG$. The Main Theorem provides a precise set of equivalent criteria for zero-dimensionality and clarifies when these imply or fail to imply each other, while also connecting these spaces to clopen bases and cozero-sets on the Yosida space. The work thus offers a complete framework for determining zero-dimensionality of maximal $d$-subgroups in W-objects and highlights the dualities with Wallman lattices and Ultrafilter representations.
Abstract
This article is a continuation of [6] where a classification of when the space of minimal prime subgroups of a given lattice-ordered group equipped with the inverse topology has a clopen $π$-base. For nice $\ell$-groups, (e.g. W-objects) this occurs precisely when the space of maximal $d$-subgroups (qua the hull kernel topology) has a clopen $π$-base. It occurred to us that presently there is no classification of when the space of maximal $d$-subgroups of a W-object is zero-dimensional, except for the case of the $C(X)$, the real-valued continuous functions on a topological space $X$, considered in [5].