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Les beaux automorphismes. (The beautiful automorphisms). (French) Zbl 0766.03022
Summary: Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models of $$T$$ which: 1. are strong, i.e. leave the algebraic closure (in $$T^{eq}$$) of the empty set pointwise fixed, 2. are obtained by the Fraïssé construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language of $$T$$ plus one unary function symbol for the automorphism), and we study this theory. In particular, we examine the question of the saturation of the beautiful automorphisms. We also prove that in some cases (in particular if the theory is $$\omega$$-stable and $$G$$-trivial), almost all (in the sense of Baire categoricity) automorphisms of the saturated countable model are beautiful and conjugate.

##### MSC:
 03C50 Models with special properties (saturated, rigid, etc.)
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##### References:
 [1] [D] Duret, J.L.: Les corps pseudo-finis ont la propriété d’indépendance. C.R. Acad. Sci. Paris, Sér. A, t.290, 981-983 (1980) · Zbl 0469.03020 [2] [L1] Lascar, D.: Autour de la propriété du petit indice. Proceedings of the London Mathematical Society,62 (3), 25-33 (1991) · Zbl 0683.03017 [3] [L2] Lascar, D.: On the category of models of a complete theory. J. Symb. Logic47, 249-266 (1982) · Zbl 0498.03019 [4] [P] Poizat, B.: Paires de structures stables. J. Symb. Logic48, 239-249 (1983) · Zbl 0525.03023 [5] [S] Shelah, S.: Stability, the fcp and superstability. Ann. Math. Logic3, 271-362 (1971) · Zbl 0281.02052
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