Quantifier elimination for o-minimal structures expanded by a valuational cut

TL;DR

This paper establishes conditions under which the theory of an o-minimal structure expanded by a valuational cut admits quantifier elimination, especially when the cut is a convex subring with an o-minimal residue field.

Contribution

It identifies a specific condition that guarantees quantifier elimination for structures expanded by a valuational cut, extending previous results in o-minimal theory.

Findings

Condition for quantifier elimination in expanded structures

Application to convex subrings with o-minimal residue fields

Theoretical framework for analyzing valuational cuts

Abstract

Let $R$ be an o-minimal expansion of a group in a language in which $\textrm{Th}(R)$ eliminates quantifiers, and let $C$ be a predicate for a valuational cut in $R$. We identify a condition that implies quantifier elimination for $\textrm{Th}(R,C)$ in the language of $R$ expanded by $C$ and a small number of constants, and which, in turn, is implied by $\textrm{Th}(R,C)$ having quantifier elimination and being universally axiomatizable. The condition applies for example in the case when $C$ is a convex subring of an o-minimal field $R$ and its residue field is o-minimal.