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

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.