I am trying to find an example of a totally ordered abelian group $(\Gamma,+)$ such that the set of its convex subgroups is $\textbf{not}$ totally ordered.
I think there should be such an example, but something tells me it is not so trivial. Any help or reference would be deeply appreciated.
It is totally ordered.
Indeed, let $A,B$ be convex subgroups, and suppose $b\in B-A$, and $a\in A-B$. Up to invert, we can suppose $a,b\ge 0$. Up to switch, we can suppose $a\le b$. So $0\le a\le b$. By convexity, $a\in B$, contradiction.
Note that I used very little of the axioms: I only need a set $G$ with a total order and a basepoint $0$, and an involution $i$ exchanging $G_{\ge 0}$ and $G_{\le 0}$: then the set of $i$-invariant convex subsets is totally ordered.