Algebra universalis

, 79:12 | Cite as

Canonical extensions of bounded archimedean vector lattices

  • Guram Bezhanishvili
  • Patrick J. Morandi
  • Bruce Olberding
Article
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Part of the following topical collections:
  1. In memory of Bjarni Jónsson

Abstract

Canonical extensions of Boolean algebras with operators were introduced in the seminal paper of Jónsson and Tarski. The two defining properties of canonical extensions are the density and compactness axioms. While the density axiom can be extended to the setting of vector lattices of continuous real-valued functions, the compactness axiom requires appropriate weakening. This provides a motivation for defining the concept of canonical extension in the category \(\varvec{ bav }\) of bounded archimedean vector lattices. We prove existence and uniqueness theorems for canonical extensions in \(\varvec{ bav }\). We show that the underlying vector lattice of the canonical extension of \(A\in \varvec{ bav }\) is isomorphic to the vector lattice of all bounded real-valued functions on the Yosida space of A, and give an intrinsic characterization of those \(B \in \varvec{ bav }\) that arise as the canonical extension of some \(A \in \varvec{ bav }\).

Keywords

Vector lattice Canonical extension Compact Hausdorff space Continuous real-valued function Functional representation 

Mathematics Subject Classification

06F20 46A40 54D30 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Guram Bezhanishvili
    • 1
  • Patrick J. Morandi
    • 1
  • Bruce Olberding
    • 1
  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

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