Kikyo, Hirotaka Model companions of theories with an automorphism. (English) Zbl 0967.03031 J. Symb. Log. 65, No. 3, 1215-1222 (2000). Some years ago, A. Macintyre, L. van den Dries and C. Wook realized that the theory of algebraically closed fields with a distinguished automorphism \(\sigma\) has a model companion (usually called ACFA) and equipped ACFA with a nice axiomatization. Later Z. Chatzidakis and E. Hrushovski studied ACFA (as a typical simple unstable theory) intensively. Turning from this particular algebraic framework to a general purely model-theoretic setting, one may consider, for every given theory \(T\), the theory \(T_{\sigma}\) of models of \(T\) with a new automorphism \(\sigma\), and wonder when \(T_{\sigma}\) has a model companion, or even a simple model companion, provided that \(T\) is simple. Chatzidakis and Pillay showed in 1998 that, if \(T\) is stable, equals \(T^{\text{eq}}\) and eliminates quantifiers, then the model companion of \(T_{\sigma}\) is simple, if it exists. But it was observed by Kudajbergenov that sometimes no model companion arises: this is the case, for instance, when \(T\) is stable and has the finite cover property. The paper under review examines this question when \(T\) is a model complete theory, in order to single out the key conditions forbidding a model companion of \(T_{\sigma}\). Under this perspective the author shows that no model companion exists when \(T\) is unstable and either \(T\) does not satisfy the independence property, or \(T_{\sigma}\) has the amalgamation property. With respect to the finite cover property, it is proved that this assumption is sufficient to exclude a model companion when \(T = T^{\text{eq}}\) eliminates quantifiers and has PAPA (the amalgamation property for local automorphisms). Several examples, including the theory \(T\) of random graphs, are discussed to illustrate these results; in particular, in the case of random graphs (another typical simple unstable setting), \(T\) has the independence property, but \(T_{\sigma}\) has no model companion. Reviewer: C.Toffalori (Camerino) Cited in 1 ReviewCited in 9 Documents MSC: 03C45 Classification theory, stability, and related concepts in model theory Keywords:theories with an automorphism; model complete theory; model companion; independence property; amalgamation property; finite cover property PDFBibTeX XMLCite \textit{H. Kikyo}, J. Symb. Log. 65, No. 3, 1215--1222 (2000; Zbl 0967.03031) Full Text: DOI arXiv References: [1] Classification theory and the number of non-isomorphic models (1978) [2] An introduction to stability theory (1983) [3] Logic Colloquium ’84 pp 121–153– (1986) [4] Annals of Pure and Applied Logic 95 pp 71–92– (1998) [5] Proceedings of the London Mathematical Society 62 pp 25–53– (1991) [6] Model theory (1993) [7] Archive for Mathematical Logic 31 pp 55–68– (1991) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.