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Bi-orders do not arise from total orders

Part of: Set theory Ordered structures

Published online by Cambridge University Press:  20 April 2021

Samuel M. Corson
Samuel M. Corson*
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, United Kingdom
*
e-mail: sammyc973@gmail.com
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Abstract

We present a Zermelo–Fraenkel ( $\textbf {ZF}$ ) consistency result regarding bi-orderability of groups. A classical consequence of the ultrafilter lemma is that a group is bi-orderable if and only if it is locally bi-orderable. We show that there exists a model of $\textbf {ZF}$ plus dependent choice in which there is a group which is locally free (ergo locally bi-orderable) and not bi-orderable, and the group can be given a total order. The model also includes a torsion-free abelian group which is not bi-orderable but can be given a total order.

Keywords

Left-orderable groupbi-orderable groupultrafilter lemma

MSC classification

Primary: 03E25: Axiom of choice and related propositions
Secondary: 06F15: Ordered groups 06F20: Ordered abelian groups, Riesz groups, ordered linear spaces
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 217 - 224
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This work was supported by the Severo Ochoa Program for Centres of Excellence in R&D SEV-20150554 and the Heilbronn Institute for Mathematical Research Bristol, UK.

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