Weakly o-minimal types
Slavko Moconja, Predrag Tanović
TL;DR
The paper develops a comprehensive framework for weakly o-minimal types in arbitrary complete theories, defining weakly o-minimality via a relative order on the locus of a type and proving that such types are dp-minimal with stable-like and forking-theoretic properties. It characterizes all relatively definable orders on a weakly o-minimal locus as iterated reversals along convex equivalence classes and establishes monotonicity theorems for relatively definable functions, both locally and globally. It also builds a robust theory of so-types, their genericity notions (left/right), and nonorthogonality relations, including direct nonorthogonality and orientation-preserving extensions, with substantial results on how these notions behave under forking and extensions. Collectively, these results extend o-minimal intuition to a broader setting, providing tools for analysis of weakly quasi-o-minimal theories and laying groundwork for applications to Vaught/Martin-type conjectures and dp-minimal model theory. The work thus offers foundational methods for understanding geometric and combinatorial structure of types in non-o-minimal contexts, particularly through monotonicity, convex partitions, and genericity concepts.
Abstract
We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every relatively $L_{\mathfrak{C}}$-definable subset of $p(\mathfrak{C})$ has finitely many convex components in $(p(\mathfrak{C}),<)$. 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.