Extending partial isomorphisms for the small index property of many \(\omega\)-categorical structures
type: Item
instance of: scholarly article
title: Extending partial isomorphisms for the small index property of many \(\omega\)-categorical structures
zbMATH Open document ID: 0922.03044
DOI: 10.1007/BF02764005
author: Bernhard Herwig
published in: Israel Journal of Mathematics
publication date: 1999-03-09T00:00:00Z
review text: For a finite \(K_{m}\)-free graph \(A\) (\(K_{m}\) is the complete graph with \(m\) vertices) and partial isomorphisms \(p_{1},\ldots, p_{n}\) on \(A\), it is proven that there are a finite extension \(B\), which is also a \(K_{m}\)-free graph, and automorphisms \(f_{i}\) of \(B\), extending the \(p_{i}\) (the property was earlier called the extension property). \textit{W. Hodges, I. Hodkinson, D. Lascar}, and \textit{S. Shelah} [J. Lond. Math. Soc., II. Ser. 48, No. 2, 204-218 (1993; Zbl 0788.03039)] shows that the property is important to understand the automorphism group of a random graph as a topological group. The property was used in the cited paper to prove the small index property (a countable structure \(M\) has the small index property if for every subgroup \(H\) of \(G={\text{Aut}}(M)\), \(H\) is open whenever \([G:H]<2^{\omega}\)) for random graphs. Analogous results are proven here for the generic countable \(K_{m}\)-free graphs and for the generic \(\omega\)-categorical digraphs introduced by \textit{C. W. Henson} [J. Symb. Log. 37, No. 3, 494-500 (1972; Zbl 0259.02040)].
Mathematics Subject Classification ID: 03C13
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C80
Mathematics Subject Classification ID: 03C30
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 1239660
zbMATH Keywords: graph
zbMATH Keywords: finite graph
zbMATH Keywords: random graph
zbMATH Keywords: \(\omega\)-categorical digraph
zbMATH Keywords: automorphism group
zbMATH Keywords: free amalgam
zbMATH Keywords: generic structure
zbMATH Keywords: extension property
zbMATH Keywords: small index property
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1239660