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Extending partial isomorphisms for the small index property of many ω-categorical structures

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Abstract

Theorem:Let A be a finite K m -free graph, p 1 , …, p n partial isomorphisms on A. Then there exists a finite extension B, which is also a K m -free graph, and automorphisms f i of B extending the p i .

A paper by Hodges, Hodkinson, Lascar and Shelah shows how this theorem can be used to prove the small index property for the generic countable graph of this class. The same method also works for a certain class of continuum many non-isomorphic ω-categorical countable digraphs and more generally for structures in an arbitrary finite relational language, which are built in a similar fashion. Hrushovski proved this theorem for the class of all finite graphs [Hr]; the proof presented here stems from his proof.

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References

  1. H. Andréka, I. Hodkinson and I. Németi,Finite algebras of relations are representable on finite sets, to appear in Journal of Symbolic Logic.

  2. E. Ahlbrandt and M. Ziegler,Quasi finitely axiomatizable totally categorical theories, Annals of Pure and Applied Logic30 (1986), 63–82.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. Cameron,Oligomorphic Permutation Groups, London Mathematical Society Lecture Notes Series 152, Cambridge University Press, 1990.

  4. R. Fraïssé,Sur l'extension aux rélations de quelques propriété des ordres, Annales Scientifiques de l'École Normale Supérieure71 (1954), 361–388.

    Google Scholar 

  5. M. Grohe,Arity hierarchies, Annals of Pure and Applied Logic82 (1996), 103–163.

    Article  MATH  MathSciNet  Google Scholar 

  6. M. Goldstern, R. Grossberg and M. Kojman,Infinite homogeneous bipartite graphs with unequal sides, Discrete Mathematics149 (1996), 69–82.

    Article  MATH  MathSciNet  Google Scholar 

  7. W. Henson,Countable homogeneous relational structures and ℵ 0 theories, The Journal of Symbolic Logic37 (1972), 494–500.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Herwig,Extending partial isomorphisms, Combinatorica15 (1995), 365–371.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Herwig and D. Lascar,Extending partial isomorphisms and the profinite topology on the free groups, to appear in Transactions of the American Mathematical Society.

  10. E. Hrushovski,Extending partial isomorphisms of graphs, Combinatorica12 (1992), 411–416.

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Hodges, I. Hodkinson, D. Lascar and S. Shelah,The small index property for ω-stable, ω-categorical structures and for the random graph, Journal of the London Mathematical Society48 (1993), 204–218.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Kaye and D. Macpherson,Automorphisms of First-Order Structures, Clarendon Press, Oxford, 1994.

    MATH  Google Scholar 

  13. D. Lascar,Autour de la propriété du petit indice, Proceedings of the London Mathematical Society62 (1991), 25–53.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. G. Peretyat'kin,On countable theories with a finite number of denumerable models, Algebra i Logika12 (1973), 570–576 (310–326 in the English translation).

    Google Scholar 

  15. J. K. Truss,Generic automorphisms of homogeneous structures, Proceedings of the London Mathematical Society65 (1992), 121–141.

    Article  MATH  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Institut für Mathematische Logik, Universität Freiburg, Eckerstr. 1, 79104, Freiburg, Germany

    Bernhard Herwig

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  1. Bernhard Herwig
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Correspondence to Bernhard Herwig.

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Supported by EC-grant ERBCHBGCT 920013.

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Herwig, B. Extending partial isomorphisms for the small index property of many ω-categorical structures. Isr. J. Math. 107, 93–123 (1998). https://doi.org/10.1007/BF02764005

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  • DOI: https://doi.org/10.1007/BF02764005

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