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

Authors
  • Darnel, Michael R.1
  • Martinez, Jorge2
  • 1 Indiana University South Bend, Department of Mathematics, South Bend, IN, 46634, U.S.A , South Bend
  • 2 University of Florida, Department of Mathematics, Gainesville, FL, 32611-8105, U.S.A , Gainesville
Type
Published Article
Journal
Order
Publisher
Kluwer Academic Publishers
Publication Date
Mar 01, 2002
Volume
19
Issue
1
Pages
35–72
Identifiers
DOI: 10.1023/A:1015259615457
Source
Springer Nature
Keywords
License
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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(discκ), where discκ 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∈GY(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(discκ). 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⊥.

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