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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2510.16967 |
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| _version_ | 1866915608346492928 |
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| author | Wang, Xiaoduo |
| author_facet | Wang, Xiaoduo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations Wang, Xiaoduo Logic 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. |
| title | T-Convexity, Tame Extensions and Definability of Hausdorff Limits in O-minimal Structures with Generic Derivations |
| topic | Logic |
| url | https://arxiv.org/abs/2510.16967 |