Definable ranks
Lothar Sebastian Krapp, Salma Kuhlmann, Lasse Vogel
TL;DR
This work introduces and analyzes the definable rank for ordered fields, ordered abelian groups, and ordered sets, linking it to the natural valuation and its induced value group and value set. It provides a full characterization of definable convex subgroups on the group level using Schmitt's spine framework and then relates field- and group-level definable ranks, especially for henselian natural valuations, showing that the ranks are tightly connected with precise cases of coincidence or controlled difference. The authors also develop a method to study definable final segments of ordered sets via a definable condensation, giving concrete results and illuminating open cases. Collectively, the results clarify when definable ranks align across field, group, and set levels and establish tools for further exploration of definability in valued-ordered structures, with implications for Shelah-type conjectures and related definability questions.
Abstract
We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly, we compare the definable rank of an ordered abelian group to that of its value set with respect to the natural valuation. We fully describe the definable rank on the group level. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and develop a tool for further study.