# Extensions of partial orders to linear orders on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order?

The answer is affirmative on abelian groups, where being torsion-free is necessary and sufficient both for having a linear order and for partial orders to extend to linear orders (Fuchs, 1950, since any partial order on a torsion-free abelian group extends to a normal order in his terminology).

• Groups in which every bi-invariant partial order extends to a bi-invariant order are called $O^*$-group in Gupta, Narain; Rhemtulla, Akbar On ordered groups. Algebra Univ. 1 (1971/72), 129-132. Possibly more information can be found in Rhemtulla's book, but I haven't yet had access to it.
– YCor
Feb 16, 2020 at 19:39

There are necessary and sufficient conditions in the literature for a (left) partial order $$\le$$ on $$G$$ to extend to a (left) linear order $$\le^{\ast}$$ on $$G$$. This shows in particular, that not every partial left order extends to a linear left order in the non-abelian case, even though the group is orderable.
• I see. I thought that not every partial left order on $G$ extends to a linear left order on $G$ in the non-abelian case. I will check again (there is Conrad's theorem, see arxiv.org/pdf/math/0606264.pdf, section $4$). Sep 17, 2013 at 18:05