Definable ranks

TL;DR

This paper introduces the concept of definable rank in ordered structures, explores its properties, and compares ranks across fields, groups, and valuations, providing new insights into their interrelations and definability aspects.

Contribution

It defines and analyzes the definable rank for ordered fields, groups, and sets, and compares these ranks through valuation theory, including for henselian valued fields.

Findings

Characterization of definable convex subgroups in ordered groups

Comparison of definable ranks between fields and their value groups

Introduction of definable condensation as a new analytical tool

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 describe the definable rank on the group-level by characterizing the definable convex subgroups. 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 introduce definable condensation as a tool for further study.