#### Vol. 3, No. 7, 2009

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Discretely ordered groups

### Peter A Linnell, Akbar Rhemtulla and Dale P. O. Rolfsen

Vol. 3 (2009), No. 7, 797–807
##### Abstract

We consider group orders and right-orders which are discrete, meaning there is a least element which is greater than the identity. We note that nonabelian free groups cannot be given discrete orders, although they do have right-orders which are discrete. More generally, we give necessary and sufficient conditions that a given orderable group can be endowed with a discrete order. In particular, every orderable group $G$ embeds in a discretely orderable group. We also consider conditions on right-orderable groups to be discretely right-orderable. Finally, we discuss a number of illustrative examples involving discrete orderability, including the Artin braid groups and Bergman’s nonlocally-indicable right orderable groups.

discrete order
##### Mathematical Subject Classification 2000
Primary: 20F60
Secondary: 06F15, 20F36
##### Milestones
Revised: 9 March 2009
Accepted: 17 April 2009
Published: 29 November 2009
##### Authors
 Peter A Linnell Department of Mathematics Virginia Tech Blacksburg, VA 24061-0123 United States http://www.math.vt.edu/people/plinnell/ Akbar Rhemtulla Department of Mathematical & Statistical Sciences University of Alberta Edmonton, AL  T6G 2G1 Canada http://www.math.ualberta.ca/Rhemtulla_A.html Dale P. O. Rolfsen University of British Columbia Mathematics Department 1984 Mathematics Road Vancouver, BC  V6T 1Z2 Canada http://www.math.ubc.ca/~rolfsen/