Generic stability, regularity, and quasiminimality

TL;DR

This paper explores the relationships between various stability and regularity notions in first-order theories, establishing equivalences and properties of quasiminimal structures and types, with implications for model-theoretic classification.

Contribution

It proves that infinite-dimensional homogeneous pregeometries coincide with generically stable strongly regular types and characterizes quasiminimal structures of size at least aleph-2 as homogeneous pregeometries.

Findings

Infinite-dimensional homogeneous pregeometries equal generically stable strongly regular types.

Quasiminimal structures of size ≥ aleph-2 are homogeneous pregeometries.

Generic type of any quasiminimal structure is locally strongly regular.

Abstract

We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in an arbitrary first order theory T. We prove that "infinite-dimensional homogeneous pregeometries" coincide with generically stable strongly regular types (p(x),x=x). We prove that quasiminimal structures of cardinality at least aleph-2 are homogeneous pregeometries, We prove that the generic type of an arbitrary quasiminimal structure is locally strongly regular. Some of the results depend on a general dichotomy for regular-like types: generic stability, or existence of a suitable definable partial ordering.