O-minimal residue fields of o-minimal fields

TL;DR

This paper characterizes exactly when the residue field of an o-minimal field with a convex subring is itself o-minimal, providing a complete axiomatization of such structures.

Contribution

It proves that the previously known sufficient conditions for o-minimality of the residue field are also necessary, completing the characterization.

Findings

Necessary and sufficient conditions for o-minimality of residue fields

Axiomatization of structures with o-minimal residue fields

Extension of previous sufficiency results to necessity

Abstract

Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures (R,V) such that k_ind, the corresponding residue field with structure induced from R via the residue map, is o-minimal. More precisely, in previous work it was shown that certain first order conditions on (R,V) are sufficient for the o-minimality of k_ind. Here we prove that these conditions are also necessary.