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

Abstract

Let R be an o-minimal expansion of a group in a language in which 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 Th(R,C) in the language of R expanded by C and a small number of constants, and which, in turn, is implied by 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.