Extending partial isomorphisms for the small index property of many ω-categorical structures |
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Authors: | Bernhard Herwig |
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Institution: | 1. Institut für Mathematische Logik, Universit?t Freiburg, Eckerstr. 1, 79104, Freiburg, Germany
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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.
Supported by EC-grant ERBCHBGCT 920013. |
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Keywords: | |
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