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Positivity

pp 1–7 | Cite as

The completeness characterization of \(C(\mathcal {L})\), \(\mathcal {L}\) a locale

  • R. N. Ball
  • A. W. Hager
Article

Abstract

The result of the title is: an archimedean \(\ell \)-group with weak unit A is (isomorphic to) \(C(\mathcal {L})\) for some (identifiable) locale \(\mathcal {L}\) (or, \(\mathbb {R}\mathcal {L}^{\mathrm {op}}\), \(\mathcal {L}^{\mathrm {op}}\) the opposite frame) iff A is divisible and “\(*\)-complete,” (a type of sequential completeness). This is from Ball and Hager (Positivity 10:165–199, 2006), and is revisited here, with streamlined proof.

Keywords

Archimedean lattice-ordered group Sequentially complete Yosida space Countable composition Locale Frame 

Mathematics Subject Classification

06D22 06F20 18A40 46A40 46E05 54A20 54B35 54C30 

Notes

Acknowledgements

We thank B. Banaschewski for numerous conversations, which spawned the present effort, and also [9]. We thank the referee for a careful reading and helpful suggestions.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DenverDenverUSA
  2. 2.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA

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