Generic stability, regularity, and quasiminimality

Details Generic stability, regularity, and quasiminimality Generic stability, regularity, and quasiminimality

2009-12-06

arxiv.org/abs/0912.1115v2

Mathematics

Logic

32 pages

Scientific paper

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.

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