Extentability of Automorphisms of Generic Substructures

2014/04/05 by Aristotelis Panagiotopoulos, Panagiotopoulos, Aristotelis
Mathematics · #03C15 #03C50 #03C52 #03E15 #51F99 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03C15 #msc:03C50 #msc:03C52 #msc:03E15 #msc:51F99

paper · pdf · doi:10.48550/arxiv.1404.1427

21 pages

arxiv created 2014/10/26 · arxiv updated 2014/10/28

Abstract

We show that if g is a generic (in the sense of Baire category) isometry of a generic subspace of the Urysohn metric space U, then g does not extend to a full isometry of U. The same holds for the Urysohn sphere S. Let M be a Fraisse L-structure, where L is a relational countable language and M has no algebraicity. We provide necessary and sufficient conditions for the following to hold: for a generic substructure A of M, every automorphism f in Aut(A) extends to a full automorphism f in Aut(M). From our analysis, a dichotomy arises and some structural results are derived that, in particular, apply to omega-stable Fraisse structures without algebraicity.

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