Ordered Algebraic Structures pp 275-299 | Cite as

# Polar Functions, I: The Summand-Inducing Hull of an Archimedean *l*-Group with Unit

## Abstract

This is an introduction to the concept of a polar function on W, the category of archimedean l-groups with designated unit, and to that of its dual, a covering function on compact spaces.

Let *X* be a subalgebra of the boolean algebra of polars *P(G)* of the W-object *G.* An essential extension *H* of *G* is said to be an *X*-splitting extension of *G* if the extension of each *K* ∈ *P(G)* to *H* is a cardinal summand. The least *X*-splitting extension *G[X]* of *G* is studied here. Dually, one considers a compact Hausdorff space, and a subalgebra *k* of *R(X)*, the boolean algebra of all regular closed sets. A *k*-cover *Y* of *X* is represented by an irreducible map *g : Y→X* subject to the condition that clyg^{-1}(int ^{X} A) is clopen, for each A ∈ k. There is a minimum k-cover. To each subalgebra *X* of polars of *G* there corresponds canonically a subalgebra k of regular closed sets of the Yosida space *YG* of *G*, in such a way that the Yosida space of *G[X]* is the minimum k-cover of *YG*. This general setup is applied to some well known situations, to recapture constructions such as the projectable hull. On the topological side one may recover the cloz cover of a compact space.

A function which assigns to each *G* a subalgebra *X(G)* of polars of *G* is called a polar function. The dual notion for compact spaces is the covering function: assigning to each space *X* a subalgebra *k(X)* of regular closed sets. By transfinitely iterating the basic constructions of least *X*-splitting extensions and minimum k-covers, one obtains their idempotent closures, *X* ^{b} and *k* ^{b}, respectively. These closures give rise to, respectively, hull classes of archimedean *l*-groups and covering classes of compact spaces.

## Keywords

Covering Function Boolean Algebra Compact Space Polar Function Compact Hausdorff Space## Preview

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## References

- [BKW77]A. Bigard, K. Keimel & S. Wolfenstein,
*Groupes et Anneaux Réticulés.*Lecture Notes in Math**608**, Springer Verlag (1977); Berlin-Heidelberg-New York.MATHGoogle Scholar - [Bl74]R. D. Bleier,
*The SP-hull of a lattice-ordered group.*Canad. Jour. Math.**XXVI**, No. 4 (1974), 866–878.MathSciNetCrossRefGoogle Scholar - [Ch71]D. Chambless,
*The Representation and Structure of Lattice-Ordered Groups an f-Rings.*Tulane University Dissertation (1971), New Orleans.Google Scholar - [C71]P. F. Conrad,
*The essential closure of an archimedean lattice-ordered group.*Duke Math. Jour.**38**(1971), 151–160.MathSciNetMATHCrossRefGoogle Scholar - [C73]P. F. Conrad,
*The hulls of representable l-groups and f-rings.*Jour. Austral. Math. Soc.**26**(1973), 385–415.MathSciNetCrossRefGoogle Scholar - [D95]M. R. Darnel,
*The Theory of Lattice-Ordered Groups.*Pure & Appl. Math.**187**, Marcel Dekker (1995); Basel-Hong Kong-New York.Google Scholar - [GJ76]L. Gillman & M. Jerison,
*Rings of Continuous Functions.*Grad. Texts in Math.**43**, Springer Verlag (1976); Berlin-Heidelberg-New York.MATHGoogle Scholar - [H89]A. W. Hager,
*Minimal covers of topological spaces.*In Papers on General Topology and Related Category Theory and Topological Algebra; Annals of the N. Y. Acad. Sci.**552**March 15, 1989, 44–59.MathSciNetCrossRefGoogle Scholar - [HM99]A. W. Hager & J. Martínez,
*Hulls for various kinds of a-completeness in archimedean lattice-ordered groups.*Order**16**(1999), 89–103.MathSciNetMATHCrossRefGoogle Scholar - [HM01]A. W. Hager & J. Martínez,
*The ring of α-quotients.*To appear, Algebra Universalis.Google Scholar - [HM∞a]A. W. Hager
*&*J. Martínez,*Polar functions, II: completion classes of archimedean f-algebras vs. covers of compact spaces.*Preprint.Google Scholar - [HM∞b]A. W. Hager & J. Martínez,
*The projectable and regular hulls of a semiprime ring.*Work in progress.Google Scholar - [HR77]A. W. Hager & L. C. Robertson,
*Representing and ringifying a Riesz space.*Symp. Math.**21**(1977), 411–431.MathSciNetGoogle Scholar - [HVW89]M. Henriksen, J. Vermeer & R. G. Woods,
*Wallman covers of compact spaces.*Diss. Math.**CCLXXX**(1989), Warsaw.Google Scholar - [MMc∞]J. Martínez & W. Wm. McGovern,
*C*-compactifications.*Ongoing research.Google Scholar - [Mc98]W. Wm. McGovern,
*Algebraic and Topological Properties of C(X) and the F-topology.*University of Florida Dissertation, 1998; Gainesville, FL.Google Scholar - [PW89]J. R. Porter & R. G. Woods,
*Extensions and Absolutes of Hausdorff Spaces.*Springer Verlag (1989); Berlin-Heidelberg-New York.Google Scholar - [V84a]J. Vermeer,
*On perfect irreducible preimages.*Topology Proc.**9**(1984), 173–189.MathSciNetMATHGoogle Scholar - [V84b]J. Vermeer,
*The smallest basically disconnected preimage of a space.*Topology and its Appl.**17**(1984), 217–232.MathSciNetCrossRefGoogle Scholar