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

TL;DR

This paper characterizes the images of cross-sections of surjective morphisms between divisible abelian o-groups, showing they are exactly the divisible, tame, and cofinal subgroups compatible with the morphism, with applications to real closed valued fields.

Contribution

It provides a complete description of cross-section images in divisible abelian o-groups and links these structures to real closed valued fields.

Findings

Images are exactly divisible, tame, and cofinal subgroups

Subgroups are compatible with the morphism in a specific sense

Application to real closed valued fields

Abstract

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