Cross-Sections of Divisible Abelian $o$-Groups via Tame Pairs

It is shown that images of cross-sections of surjective morphisms $f:\mathrm{\Gamma }⟶\mathrm{\Delta }$ of divisible abelian $o$-groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of $\mathrm{\Gamma }$ compatible with $f$ in a suitable sense. The note concludes with an application to real closed valued fields.