O-minimal fields with standard part map

TL;DR

This paper studies the structure of definable sets in o-minimal fields with a standard part map, identifying conditions for o-minimality of the induced structure and characterizing definable sets in specific saturated cases.

Contribution

It establishes conditions under which the induced structure on the residue field remains o-minimal and characterizes definable sets in omega-saturated cases with a specific convex subring.

Findings

Conditions for o-minimality of the induced structure

Characterization of definable sets in omega-saturated cases

Standard parts of definable sets correspond to definable sets in the residue field

Abstract

Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of k_{ind}. We also show that if R is omega-saturated and V is the convex hull of the rationals in R, then the sets definable in k_{ind} are exactly the standard parts of the sets definable in (R,V).