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Dense Orderings in the Space of Left-orderings of a Group

Part of: Ordered structures

Published online by Cambridge University Press:  17 December 2019

Adam Clay  and
Tessa Reimer
Adam Clay
Affiliation:
Department of Mathematics, 420 Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2 Email: Adam.Clay@umanitoba.careimer46@myumanitoba.ca URL: http://server.math.umanitoba.ca/∼claya/
Tessa Reimer
Affiliation:
Department of Mathematics, 420 Machray Hall, University of Manitoba, Winnipeg, MB, R3T 2N2 Email: Adam.Clay@umanitoba.careimer46@myumanitoba.ca URL: http://server.math.umanitoba.ca/∼claya/
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Abstract

Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally partitioned into two subsets. This note investigates the structure of this partition, specifically the set of dense orderings of a group and its closure within the space of orderings. We show that for bi-orderable groups, this closure will always contain the space of Conradian orderings—and often much more. In particular, the closure of the set of dense orderings of the free group is the entire space of left-orderings.

Keywords

ordered groupspace of left-orderingsbi-orderable group

MSC classification

Primary: 06F15: Ordered groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 393 - 404
Copyright
© Canadian Mathematical Society 2019

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Footnotes

Adam Clay was partially supported by NSERC grant RGPIN-2014-05465.

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