Stable generic structures
type: Item
instance of: scholarly article
title: Stable generic structures
zbMATH Open document ID: 0857.03020
DOI: 10.1016/0168-0072(95)00027-5
author: Niandong Shi
author: John T. Baldwin
published in: Annals of Pure and Applied Logic
publication date: 1997-03-11T00:00:00Z
review text: The aim of the paper is to exhibit a set of axioms which for collections of finite structures with dimension function give rise to stable generic models. The paper includes the investigation of an almost strongly minimal non-Desarguesian projective plane as well as the development of the new case where finite sets may have infinite closures with respect to the dimension function. The investigated generic structures need not be \(\omega\)-saturated, are flat, and do not interpret a group. In Section 1 the authors give an overview of the contents and connections with constructions in other papers. Section 2 is busy with the definition of a generic model having interesting properties from a collection of finite structures. Further, the authors present four examples which show that slight variations on the notion of strong substructure yield generic models with quite different properties. In the next section a dimension function \(d\) on finite structures is defined. This concept allows to define the notion of strong submodel and a notion of free amalgamation. The authors prove some sufficient conditions for the stability of generic models and for the coincidence of \(d\)-independence with nonforking. Section 4 deals with amalgamation over closed sets. The full amalgamation property implies the uniform amalgamation property. The \(({\mathbf K}_0, \leq)\)-generic model \(M\) is full, if \(({\mathbf K}_0, \leq)\) has the full amalgamation property \(({\mathbf K}_0\) is a class of finite structures and \(\leq\) is a refinement of the substructure relation). In Section 5 the authors show the noninterpretability of infinite groups in the investigated structures. Up to here, the basis of consideration was an arbitrary finite relational language, while in Section 6 the authors restrict themselves to the study of graph classes. Defining the notion of a pseudoplane and the notion of a projective plane, they derive pseudoplanes from certain graphs, determine which classes give ri…
Mathematics Subject Classification ID: 03C68
Mathematics Subject Classification ID: 03C13
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 08A02
Mathematics Subject Classification ID: 51A35
zbMATH DE Number: 908465
zbMATH Keywords: saturated model
zbMATH Keywords: nonforking extension
zbMATH Keywords: interpretability of groups
zbMATH Keywords: finite structures
zbMATH Keywords: dimension function
zbMATH Keywords: stable generic models
zbMATH Keywords: strongly minimal non-Desarguesian projective plane
zbMATH Keywords: strong submodel
zbMATH Keywords: amalgamation
zbMATH Keywords: stability
zbMATH Keywords: infinite groups
zbMATH Keywords: graph classes
zbMATH Keywords: pseudoplane
reviewed by: Hans-Jürgen Vogel
MaRDI profile type: Publication
full work available at URL: 0168-0072(95)00027-5
OpenAlex ID: W1971005364
cites work: Some Projective Planes of Lenz-Barlotti Class I
cites work: Q3994342
cites work: An Almost Strongly Minimal Non-Desarguesian Projective Plane
cites work: K‐generic Projective Planes have Morley Rank Two or Infinity
cites work: Randomness and semigenericity
cites work: The primal framework. I
cites work: \(\aleph _ 0\)-categorical, \(\aleph _ 0\)-stable structures
cites work: Graphs omitting a finite set of cycles
cites work: Q4044558
cites work: A group in a group
cites work: Q5619083
cites work: Q4207889
cites work: Countable homogeneous relational structures and ℵ0-categorical theories
cites work: A new strongly minimal set
cites work: On generic structures
cites work: Q3778001
cites work: Zero-One Laws for Sparse Random Graphs
cites work: Q3037409
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description: scientific article; zbMATH DE number 908465