Know Your Rank!

Abstract

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

9 pages. Answering question asked in arXiv:2506.00443