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Canonical extensions of bounded archimedean vector lattices

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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 \(A \in \varvec{ bav }\).

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Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, 88003, USA

    Guram Bezhanishvili, Patrick J. Morandi & Bruce Olberding

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  1. Guram Bezhanishvili
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  2. Patrick J. Morandi
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  3. Bruce Olberding
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Correspondence to Guram Bezhanishvili.

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Dedicated to the memory of Bjarni Jónsson.

Presented by J. B. Nation.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J. B. Nation.

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Bezhanishvili, G., Morandi, P.J. & Olberding, B. Canonical extensions of bounded archimedean vector lattices. Algebra Univers. 79, 12 (2018). https://doi.org/10.1007/s00012-018-0495-y

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  • DOI: https://doi.org/10.1007/s00012-018-0495-y

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