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

Author:

Walsberg Erik

Abstract

Let R \mathscr {R} be an N I P \mathrm {NIP} expansion of ( R , > , + ) (\mathbb {R},>,+) by closed subsets of R n \mathbb {R}^n and continuous functions f : R m R n f : \mathbb {R}^m \to \mathbb {R}^n . Then R \mathscr {R} is generically locally o-minimal. This follows from a more general theorem on N I P \mathrm {NIP} expansions of locally compact groups, which itself follows from a result on quotients of definable sets in 1 \aleph _1 -saturated N I P \mathrm {NIP} structures by equivalence relations which are both externally definable and \bigwedge -definable. We also show that R \mathscr {R} is strongly dependent if and only if R \mathscr {R} is either o-minimal or ( R , > , + , α Z ) (\mathbb {R},>,+,\alpha \mathbb {Z}) -minimal for some α > 0 \alpha > 0 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

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