T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations

Abstract

We study the combination of two o-minimal extensions of the theory of real closed fields: one by a T-convex subring and the other by a T-derivation. Let T be a complete, model complete o-minimal extension of RCF. We show that the combined theory T_convex^delta has a model completion T_g,convex^delta. By adding a definable unary function st, we obtain a relative quantifier elimination result for tame pairs (M, delta^M, st^M, N, delta^N, st^N), where st is the standard part map and N is Dedekind complete in M. As an application, we prove the stable embedding property for tame pairs of T_g^delta. We also associate a sequence of definable metric topologies with models of T_g^delta and prove the Marker-Steinhorn Theorem for T_g^delta. As a consequence, Hausdorff limits of definable families are definable. A special case of our framework recovers Borotta's results on CODF with convex valuation subrings and tame pairs.