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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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