Know Your Rank!

TL;DR

This paper constructs specific ordered algebraic structures with matching definable ranks, answering a question about the relationship between linear orders, abelian groups, and fields.

Contribution

It provides a method to realize ordered fields and groups with prescribed definable ranks matching a given linear order, solving an open problem.

Findings

Constructed ordered abelian groups with a given archimedian spine

Built ordered fields with value groups isomorphic to these groups

Established isomorphism of definable ranks among the structures

Abstract

We study definable ranks of ordered fields, ordered abelian groups, and linear orders. For an arbitrary linear order $\Gamma$, we construct an ordered abelian group $G$ with archimedian spine $\Gamma$ and an ordered field $K$ with natural value group $G$ such that the definable ranks of $K$, $G$ and $\Gamma$ are all isomorphic. This answers a question of Krapp, Kuhlmann, and the second author.