Generic stability, regularity, and quasiminimality

We study (and sometimes introduce) the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly regular types (p(x), x = x). We prove that in a theory without the strict order property, regular types are generically stable. We give conditions under which a quasiminimal structure M is naturally a pregeometry, an example being when |M | ≥ א2.