Discretely ordered groups

Linnell, Peter A.; Rhemtulla, Akbar H.; Rolfsen, Dale P. O.

Discretely ordered groups

Authors

Linnell, Peter A.

Rhemtulla, Akbar H.

Rolfsen, Dale P. O.

Description

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 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 non-locally-indicable right orderable groups.
9 pages, minor changes, to appear in Algebra and Number theory