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Ribenboim’s order extension theorem from a constructive point of view

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Abstract

Ribenboim’s order extension theorem carries over Levi’s time-honoured result on the orderability of Abelian groups to modules over strictly ordered rings. We describe a system of generators for a distributive lattice which encodes the ideal concept of orderability. A criterion as to when this lattice collapses provides the constructive counterpart of Ribenboim’s theorem.

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Acknowledgements

The authors are grateful to the referee for expertly remarks, bibliographical hints on (the revival of) Lorenzen’s work, and generous cues for future work.

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Authors and Affiliations

  1. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Roberta Bonacina

  2. Dipartimento di Informatica, Università degli Studi di Verona, Strada le Grazie 15, 37134, Verona, Italy

    Daniel Wessel

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  1. Roberta Bonacina
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  2. Daniel Wessel
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Correspondence to Daniel Wessel.

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Daniel Wessel acknowledges financial support through the project “Categorical localisation: methods and foundations” (CATLOC) funded by the Università degli Studi di Verona within the programme “Ricerca di Base 2015”, as well as the project “A New Dawn of Intuitionism: Mathematical and Philosophical Advances” (ID 60842) funded by the John Templeton Foundation. The opinions expressed in this note are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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Bonacina, R., Wessel, D. Ribenboim’s order extension theorem from a constructive point of view. Algebra Univers. 81, 5 (2020). https://doi.org/10.1007/s00012-019-0634-0

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  • DOI: https://doi.org/10.1007/s00012-019-0634-0

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