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Generic structures and simple theories. (English) Zbl 0929.03043
The main concern of this paper is to develop a theory of generic structures for simple theories in the spirit of A. Robinson. First the authors consider the existence of structures with a generic $$n$$-ary predicate. Let $$T$$ be a complete theory in the language $$\mathcal L$$ which admits quantifier elimination. Adjoin a new $$n$$-ary predicate symbol $$P$$ to obtain the language $$\mathcal L_P$$. If the theory $$T$$ eliminates the quantifier $$\exists^{\infty}$$ (“there exist infinitely many”), $$T$$ has a model companion $$T_P$$ in the language $$\mathcal L_P$$. Then it is possible to explicitly state a set of axioms for $$T_P$$. Moreover, if $$T$$ is simple, then so is $$T_P$$. Next the language is augmented with a new symbol $$\sigma$$ to denote an automorphism. The theory $$T_0$$ is formed by adding axioms to $$T$$ saying that $$\sigma$$ is an automorphism. A criterion for the existence of a model companion $$T_A$$ of the theory $$T_0$$ is given. If $$T$$ is stable and the model companion $$T_A$$ exists, then $$T_A$$ is simple. Finally, the authors extend a result of Hrushovski’s about PAC fields [E. Hrushovski, “Pseudo-finite fields and related structures”, manuscript (1991)]. It is shown that the assumption of perfectness can be dropped in Hrushovski’s result that any complete theory of a perfect PAC field with small Galois group is simple.

##### MSC:
 03C50 Models with special properties (saturated, rigid, etc.) 03C45 Classification theory, stability, and related concepts in model theory 03C60 Model-theoretic algebra 12L12 Model theory of fields
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