Generic structures and simple theories
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 [\textit{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.
scientific article; zbMATH DE number 1308092
scientific article