Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a [total ordering](https://en.wikipedia.org/wiki/Total_order) (not to be confused with a well-ordering) which is "bi-translation invariant": a < b should imply cad < cbd.


Does anyone know any examples? 


Totally ordered *abelian* groups are easy to come up with: any direct product of subgroups of the reals, with the lexicographic ordering, will do. Knowing some non-abelian ones would help reveal what aspects of totally ordered abelian groups really depend on them being abelian...


**Edit:** Via Andy Putman's answer below, I found this great summary of results about ordered and bi-ordered groups (i.e. groups with bi-translation invariant orderings) on Dale Rolfsen's site:


[Lecture notes on Ordered Groups and Topology](http://www.math.ubc.ca/~rolfsen/papers/luminynotes/lum.pdf)


He shows numerous examples of non-abelian bi-orderable groups, including a bi-ordering (bi-translation invariant ordering) on the free group with two generators. As well, he mentions, due to Rhemtulla, that a left-orderable group is abelian iff every left-ordering is a bi-ordering, which I think really highlights the relationship between ordering and abelianity.



What's a non-abelian totally ordered group?
This concept is usually called biorderability (there is also left- and right-orderability). There are many examples, such as free groups and surface groups. Most spectacularly, the pure braid groups are biorderable, while the full braid groups are left orderable but not biorderable. The left ordering on the braid groups is usually attributed to Dehornoy, though it was discovered even earlier by Thurston (but not published).


Dale Rolfsen has several nice surveys of material related to this on his webpage [here](http://www.math.ubc.ca/~rolfsen/reprints.html). In particular, there is the complete text of a nice book called "Why are braids orderable?" that he wrote with Patrick Dehornoy, Ivan Dynnikov, and Bert Wiest. I believe that a new and much expanded edition of this book was just published.


EDIT 1 : I just found the website for the much-expanded version of Rolfsen et al's book [here](https://bookstore.ams.org/surv-148).


EDIT 2 : Thurston's construction of a left-ordering on the braid groups (which, of course, uses hyperbolic geometry) is very beautiful. It is explained very nicely in the first few pages of the paper "Orderings of mapping class groups after Thurston" by Short and Wiest, which is available on the arXiv [here](https://arxiv.org/abs/math/9907104). The intro sections of this paper also contain a brief but enlightening account of the general theory of group orderings.


Also, I have not read it, but there is a book entitled "Orderable Groups" by Rehmtulla and Mura. However, it is from 1977 and will thus omit a lot of recent work.


