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

2025/10/19 by Wang, Xiaoduo
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2510.16967

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.

Citations

• The Marker-Steinhorn Theorem
• A Guide to NIP theories
• Externally definable sets and dependent pairs
• On uniform definability of types over finite sets

Related

• Model completeness of o-minimal fields with convex valuations
• Strongly Minimal Relics of T-convex Fields
• Approximation by O-minimal sets in power-bounded T-convex valued fields
• T-convexity, Weakly Immediate Types, and T-λ-Spherical Completions of o-minimal Structures
• O-minimal fields with standard part map