Wild theories with o-minimal open core

Abstract

Abstract. Let T be a consistent o-minimal theory extending the theory of densely ordered groups and let T ′ be a consistent theory. Then there is a complete theory T * extending T such that T is an open core of T * , but every model of T * interprets a model of T ′ . If T ′ is NIP, T * can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.

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

“…Proposition 10.22 handles the positive characteristic case of Conjecture 5. This proposition was observed for noiseless NIP expansions in [39], we include a proof for the sake of completeness.…”

“…We refer to Fact 10.1, proven in [39, Proposition 6.3], as "definable selection". Definable selection can fail for noisey NIP expansions, one counterexample described in [39] is (R, <, +, Q).…”

“…Let F be an NIP (strongly dependent) field of cardinality ≤ 2 ℵ0 . Then there is a noisey NIP (strongly dependent) expansion R of (R, <, +) such that R interprets F and R • is interdefinable with (R, <, +) [39]. (Note that an expansion R of (R, <, +) is field-type if and only if R • is field type.)…”

Generalizing a theorem of Bès and Choffrut

“…(1) The primitive relations of S are boolean combinations of closed sets, the primitive functions of S are continuous, and S is strongly dependent, (2) S is either o-minimal or interdefinable with (R, αZ) for some real number α > 0 and o-minimal R with no poles and rational global scalars. In contrast arbitrary strongly dependent expansions of (R, <, +) are as complicated as arbitrary strongly dependent structures of cardinality continuum by [12].…”

Distal and non‐distal behavior in pairs

“…Note by [, § 6], for the theory of algebraically closed fields of characteristic p is NIP, but does not admit a distal expansion. By , there are even NIP expansions of the real field that do not admit a distal expansion.…”