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.

Abstract We let R be an o-minimal expansion of a field, V a convex subring, and (R0,V0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R0 is model complete relative to quantifier elimination in R, provided that kR (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of kR implies that the sets definable in kR 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).

Abstract Let M be a big o-minimal structure and G a type-definable group in M n . We show that G is a type-definable subset of a definable manifold in M n that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some M k with the topology induced by M k . Part of this result holds for the wider class of so-called invariant groups: each invariant group G in M n has a unique topology making it a topological group and inducing the same topology on a large invariant subset of the group as M n .