# zbMATH — the first resource for mathematics

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.

##### MSC:
 03C45 Classification theory, stability, and related concepts in model theory 12L12 Model theory of fields
Full Text:
##### References:
 [1] Paires de structures stables 49 pp 239– (1984) [2] Properties of forking in {$$\omega$$}-free pseudo-algebraically closed fields 67 pp 957– (2002) · Zbl 1032.03033 [3] Field arithmetic (2005) [4] DOI: 10.1090/S0273-0979-1981-14872-2 · Zbl 0466.12017 [5] DOI: 10.1016/S0168-0072(98)00021-9 · Zbl 0929.03043 [6] Journal für die Reine und Angewandte Mathematik 427 pp 107– (1992) [7] DOI: 10.1305/ndjfl/1063372196 · Zbl 1034.03040 [8] Annals of Mathematics 88 pp 239– (1967) [9] Model theory and applications 11 (2003) [10] Geometric stability theory (1996) [11] DOI: 10.1006/jabr.1997.7359 · Zbl 0922.12006 [12] Illinois Journal of Mathematics 48 pp 1321– (2004) [13] The model theory of fields 5 (1996) [14] Mathematical Proceedings 6 pp 1– (2003) [15] DOI: 10.1016/S0168-0072(97)00019-5 · Zbl 0897.03036 [16] DOI: 10.1112/S0024610798005985 · Zbl 0922.03048 [17] DOI: 10.1016/S0168-0072(00)00024-5 · Zbl 0967.03030 [18] Model theory and applications 11 (2003) [19] Une théorie de Galois imaginaire 48 pp 1151– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.