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Classification of δ-invariant amalgamation classes

Published online by Cambridge University Press:  12 March 2014

Roman D. Aref'ev ,
John T. Baldwin  and
Marco Mazzucco
Roman D. Aref'ev
Affiliation:
Informatics and Control Problems Institute, National Academy of Sciences, Almaty, Kazakstan
John T. Baldwin
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois At Chicago, Chicago, Illinois, USA, E-mail: jbaldwin@uic.edu
Marco Mazzucco
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinois At Chicago, Chicago. Illinois., USA, E-mail: marco@lac.uic.edu
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Extract

Hrushovski's generalization of the Fraisse construction has provided a rich source of examples in model theory, model theoretic algebra and random graph theory. The construction assigns to a dimension function δ and a class K of finite (finitely generated) models a countable ‘generic’ structure. We investigate here some of the simplest possible cases of this construction. The class K will be a class of finite graphs; the dimension, δ(A), of a finite graph A will be the cardinality of A minus the number of edges of A. Finally and significantly we restrict to classes which are δ-invariant. A class of finite graphs is δ-invariant if membership of a graph in the class is determined (as specified below) by the dimension and cardinality of the graph, and dimension and cardinality of all its subgraphs. Note that a generic graph constructed as in Hrushovski's example of a new strongly minimal set does not arise from a δ-invariant class.

We show there are countably many δ-invariant (strong) amalgamation classes of finite graphs which are closed under subgraph and describe the countable generic models for these classes. This analysis provides ω-stable generic graphs with an array of saturation and model completeness properties which belies the similarity of their construction. In particular, we answer a question of Baizhanov (unpublished) and Baldwin [5] and show that this construction can yield an ω-stable generic which is not saturated. Further, we exhibit some ω-stable generic graphs that are not model complete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1743 - 1750
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

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