Externally definable quotients and NIP expansions of the real ordered\n additive group

2019/10/23 by Erik Walsberg, Walsberg, Erik
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1910.10572

openalex publication_date 2019/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \R be an \NIP expansion of (\ℝ,<,+) by\nclosed subsets of \ℝⁿ and continuous functions f : \ℝ^m \→\n\ℝⁿ. Then \R is generically locally o-minimal. It follows\nthat if X \⊆ \ℝⁿ is definable in \R then the\nC^k-points of X are dense in X for any k \≥ 0. This follows from a\nmore general theorem on \NIP expansions of locally compact groups,\nwhich itself follows from a result on quotients of definable sets by\nequivalence relations which are externally definable and bigwedge-definable.\nWe also show that \R is strongly dependent if and only if\n\R is either o-minimal or\n(\ℝ,<,+,\α\ℤ)-minimal for some \α > 0.\n

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• Definable groups as homomorphic images of semilinear and field-definable groups
• Functions definable in definably complete uniformly locally o-minimal structure of the second kind
• Definable quotients in locally o-minimal structures
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