Weakly o-minimal types

Abstract

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type p ∈ S(A) is weakly o-minimal if for some relatively A-definable linear order, <, on p(C) every relatively LC-definable subset of p(C) has finitely many convex components in (p(C), <). We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.