Geodesic
Bi-orders do not arise from total orders
Canadian mathematical bulletin, Tome 65 (2022) no. 1, pp. 217-224

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DOI

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.
DOI : 10.4153/S0008439521000229
Mots-clés : Left-orderable group, bi-orderable group, ultrafilter lemma 
Corson, Samuel M. Bi-orders do not arise from total orders. Canadian mathematical bulletin, Tome 65 (2022) no. 1, pp. 217-224. doi: 10.4153/S0008439521000229
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     title = {Bi-orders do not arise from total orders},
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     year = {2022},
     volume = {65},
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     doi = {10.4153/S0008439521000229},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439521000229/}
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