Model completeness of o-minimal fields with convex valuations

TL;DR

This paper proves that o-minimality of the residue field ensures model completeness of o-minimal fields with convex valuations, extending understanding of definability and elementary extensions in such structures.

Contribution

It establishes model completeness for o-minimal fields with convex valuations under certain conditions on the residue field, and provides criteria for elementary extensions.

Findings

Model completeness holds when the residue field is o-minimal.

Definable sets in the residue field match those induced from the larger structure.

A criterion for elementary extensions of (R,V) is provided.

Abstract

We let R be an o-minimal expansion of a field, V a convex subring, and $(R_0, V_{0})$ an elementary substructure of (R,V). We let L be the language consisting of a language for R, in which R has elimination of quantifiers, and a predicate for V, and we let $L_{R_{0}}$ be the language L expanded by constants for all elements of $R_0$. Our main result is that (R,V) considered as an $L_{R_{0}}$-structure is model complete provided that $k_R$, the corresponding residue field with structure induced from R, is o-minimal. Along the way we show that o-minimality of $k_R$ implies that the sets definable in $k_R$ are the same as the sets definable in k with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).