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

  • Jorge Martínez
Part of the Developments in Mathematics book series (DEVM, volume 7)


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 KP(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.


Covering Function Boolean Algebra Compact Space Polar Function Compact Hausdorff Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BKW77]
    A. Bigard, K. Keimel & S. Wolfenstein, Groupes et Anneaux Réticulés. Lecture Notes in Math 608, Springer Verlag (1977); Berlin-Heidelberg-New York.zbMATHGoogle Scholar
  2. [Bl74]
    R. D. Bleier, The SP-hull of a lattice-ordered group. Canad. Jour. Math. XXVI, No. 4 (1974), 866–878.MathSciNetCrossRefGoogle Scholar
  3. [Ch71]
    D. Chambless, The Representation and Structure of Lattice-Ordered Groups an f-Rings. Tulane University Dissertation (1971), New Orleans.Google Scholar
  4. [C71]
    P. F. Conrad, The essential closure of an archimedean lattice-ordered group. Duke Math. Jour. 38 (1971), 151–160.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [C73]
    P. F. Conrad, The hulls of representable l-groups and f-rings. Jour. Austral. Math. Soc. 26 (1973), 385–415.MathSciNetCrossRefGoogle Scholar
  6. [D95]
    M. R. Darnel, The Theory of Lattice-Ordered Groups. Pure & Appl. Math. 187, Marcel Dekker (1995); Basel-Hong Kong-New York.Google Scholar
  7. [GJ76]
    L. Gillman & M. Jerison, Rings of Continuous Functions. Grad. Texts in Math. 43, Springer Verlag (1976); Berlin-Heidelberg-New York.zbMATHGoogle Scholar
  8. [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
  9. [HM99]
    A. W. Hager & J. Martínez, Hulls for various kinds of a-completeness in archimedean lattice-ordered groups. Order 16 (1999), 89–103.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [HM01]
    A. W. Hager & J. Martínez, The ring of α-quotients. To appear, Algebra Universalis.Google Scholar
  11. [HM∞a]
    A. W. Hager & J. Martínez, Polar functions, II: completion classes of archimedean f-algebras vs. covers of compact spaces. Preprint.Google Scholar
  12. [HM∞b]
    A. W. Hager & J. Martínez, The projectable and regular hulls of a semiprime ring. Work in progress.Google Scholar
  13. [HR77]
    A. W. Hager & L. C. Robertson, Representing and ringifying a Riesz space. Symp. Math. 21 (1977), 411–431.MathSciNetGoogle Scholar
  14. [HVW89]
    M. Henriksen, J. Vermeer & R. G. Woods, Wallman covers of compact spaces. Diss. Math. CCLXXX (1989), Warsaw.Google Scholar
  15. [MMc∞]
    J. Martínez & W. Wm. McGovern, C*-compactifications. Ongoing research.Google Scholar
  16. [Mc98]
    W. Wm. McGovern, Algebraic and Topological Properties of C(X) and the F-topology. University of Florida Dissertation, 1998; Gainesville, FL.Google Scholar
  17. [PW89]
    J. R. Porter & R. G. Woods, Extensions and Absolutes of Hausdorff Spaces. Springer Verlag (1989); Berlin-Heidelberg-New York.Google Scholar
  18. [V84a]
    J. Vermeer, On perfect irreducible preimages. Topology Proc. 9 (1984), 173–189.MathSciNetzbMATHGoogle Scholar
  19. [V84b]
    J. Vermeer, The smallest basically disconnected preimage of a space. Topology and its Appl. 17 (1984), 217–232.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Jorge Martínez
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations