## Algebra & Number Theory

### Discretely ordered groups

#### 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.

#### Article information

Source
Algebra Number Theory, Volume 3, Number 7 (2009), 797-807.

Dates
Revised: 9 March 2009
Accepted: 17 April 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797482

Digital Object Identifier
doi:10.2140/ant.2009.3.797

Mathematical Reviews number (MathSciNet)
MR2579395

Zentralblatt MATH identifier
1229.06008

Keywords
discrete order

#### Citation

Linnell, Peter; Rhemtulla, Akbar; Rolfsen, Dale. Discretely ordered groups. Algebra Number Theory 3 (2009), no. 7, 797--807. doi:10.2140/ant.2009.3.797. https://projecteuclid.org/euclid.ant/1513797482

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