×

zbMATH — the first resource for mathematics

Model companions of \(T_{\operatorname{Aut}}\) for stable \(T\). (English) Zbl 1034.03040
Summary: We introduce the notion \(T\) does not admit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property (nfcp). For any theory \(T\), form a new theory \(T_{\text{Aut}}\) by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If \(T\) is a stable theory, \(T\) does not admit obstructions if and only if \(T_{\text{Aut}}\) has a model companion. The proof involves some interesting new consequences of the nfcp.

MSC:
03C45 Classification theory, stability, and related concepts in model theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baldwin, J. T., Fundamentals of Stability Theory , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1988. · Zbl 0685.03024
[2] Chatzidakis, Z., and A. Pillay, ”Generic structures and simple theories”, Annals of Pure and Applied Logic , vol. 95 (1998), pp. 71–92. · Zbl 0929.03043
[3] Chatzidakis, Z., and U. Hrushovski, ”The model theory of difference fields”, Transactions of AMS , vol. 351 (1999), pp. 2997–3071. · Zbl 0922.03054
[4] Kikyo, H., ”Model companions of theories with an automorphism”, The Journal of Symbolic Logic , vol. 65 (2000), pp. 1215–22. JSTOR: · Zbl 0967.03031
[5] Kikyo, H., and A. Pillay, ”The definable multiplicity property and generic automorphisms”, Annals of Pure and Applied Logic , vol. 106 (2000), pp. 263–73. · Zbl 0967.03030
[6] Kikyo, H., and S. Shelah, ”The strict order property and generic automorphisms”, The Journal of Symbolic Logic , vol. 67 (2002), pp. 214–16. · Zbl 1002.03025
[7] Lascar, D., ”Les beaux automorphismes”, Archive for Mathematical Logic , vol. 31 (1991), pp. 55–68. · Zbl 0766.03022
[8] Pillay, A., ”Notes on model companions of stable theories with an automorphism”, http://www.math.uiuc.edu/People/pillay.html, 2001.
[9] Shelah, S., Classification Theory and the Number of Nonisomorphic Models , 2d edition, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990. · Zbl 0713.03013
[10] Shelah, S., ”On model completion of \(T_\mathrm aut\)”, preprint, 2003. · Zbl 0453.03040
[11] Winkler, P., ”Model completeness and Skolem expansions”, pp. 408–64 in Model Theory and Algebra: A Memorial Tribute to Abraham Robinson , edited by D. H. Saracino and V. B. Weispfenning, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975. · Zbl 0307.00005
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.