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On PAC and bounded substructures of a stable structure. (English) Zbl 1100.03027
Summary: We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing work of E. Hrushovski [“Pseudo-finite fields and related structures”, Quad. Mat. 11, 151–212 (2002; Zbl 1082.03035)]. We give precise definitions and equivalences, saying what it means for properties such is PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a “descent theorem” for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically closed fields from G. Cherlin, L. van den Dries and A. Macintyre’s preprint [The elementary theory of regularly closed fields (1980)] are also valid for pseudo-differentially closed fields.

03C45 Classification theory, stability, and related concepts in model theory
12L12 Model theory of fields
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