The aim of this note is to determine whether certain non-o-minimal expansions of o-minimal theories which are known to be NIP, are also distal. We observe that while tame pairs of o-minimal structures and the real field with a discrete multiplicative subgroup have distal theories, dense pairs of o-minimal structures and related examples do not.
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.
The aim of this work is an analysis of distal and non-distal behavior in dense pairs of o-minimal structures. A characterization of distal types is given through orthogonality to a generic type in M eq , non-distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.Definition 2.1 Let M |= T , s = (s 1 , . . . , s n ) and t = (t 1 , . . . , t m ) be tuples of sorts, and A ⊆ M s be definable. Then for any definable or type-definable S ⊆ M t , we say S is A-small if there is ∈ N and a definable f :We now remind the reader of certain definitions involving types that appear often in the context of NIP theories. Definition 2.2 Let t = (t 1 , . . . , t n ) be a tuple of sorts, and p(x) ∈ S t (U t ). We say p(x) is generically stable if there is a small A ⊂ U such that
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