Generic stability, regularity and quasiminimality

Abstract

We study notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitary first order theories. We prove that "infinite-dimensional homogeneous pregeometries" coincide with generically stable strongly regular types (p(x), x=x). We prove quasiminimal structures of cardinality at least N2 are "homogeneous pregeometries." We prove that the "generis type" of an arbitary quasiminimal structure is "locally strongly regular." Some of the results depend on a general dichotomy for "regular-like" types: generic stability, or the existence of a suitable definable partial ordering.