Stable generic structures
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 rise to projective and pseudoplanes, and consider the stability class of the associated structures.
strongly minimal non-Desarguesian projective plane
scientific article; zbMATH DE number 908465
scientific article