Generic structures and simple theories
type: Item
instance of: scholarly article
title: Generic structures and simple theories
zbMATH Open document ID: 0929.03043
DOI: 10.1016/S0168-0072(98)00021-9
published in: Annals of Pure and Applied Logic
publication date: 2000-01-05T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03C50
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 1308092
zbMATH Keywords: simple theory
zbMATH Keywords: generic structure
zbMATH Keywords: model companion
zbMATH Keywords: generic automorphism
zbMATH Keywords: PAC field
author: Zoé Chatzidakis
author: Anand Pillay
MaRDI profile type: Publication
cites work: Model theory of difference fields
cites work: Idéaux et types sur les corps séparablement clos
cites work: Q5534353
cites work: Q4830003
cites work: Forking in Simple Unstable Theories
cites work: Simple theories
cites work: Q3801533
cites work: Les beaux automorphismes. (The beautiful automorphisms)
cites work: Some remarks on definable equivalence relations in O-minimal structures
cites work: Corps et chirurgie
cites work: Q4039792
cites work: Simple unstable theories
cites work: Notes on the stability of separably closed fields
description: scientific article; zbMATH DE number 1308092