Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

Abstract

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of ℓ-groups G such that for each g∈G, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ℓ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some g∈G∈R. The correspondences T↦Y(T) and R↦T(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of ℓ-groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable ℓ-groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal κ, the class Y(disc_κ), where disc_κ stands for the class of all compact κ-disconnected spaces. Sample results follow. Every strongly projectable ℓ-group lies in Y(e.d.). The ℓ-group G lies in Y(e.d.) if and only if for each g∈G Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc_κ). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean ℓ-group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P ^⊥+Q ^⊥.