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COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS, FREE AMALGAMATION AND AUTOMORPHISM GROUPS

Published online by Cambridge University Press:  06 May 2019

DAOUD SINIORA  and
SŁAWOMIR SOLECKI
DAOUD SINIORA
Affiliation:
DEPARTMENT OF MATHEMATICS AND ACTUARIAL SCIENCE THE AMERICAN UNIVERSITY IN CAIRO AUC AVENUE, PO BOX 74 NEW CAIRO 11835, EGYPT E-mail: daoud.siniora@aucegypt.edu
SŁAWOMIR SOLECKI
Affiliation:
DEPARTMENT OF MATHEMATICS CORNELL UNIVERSITY 310 MALOTT HALL ITHACA, NY14853, USA E-mail: ssolecki@cornell.edu
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Abstract

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all ${\cal F}$-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.

Keywords

Primary 03C1303C1522F2020B27Secondary 08A3554H11extension property for partial automorphismscoherent EPPAFraïssé limitsample genericshomogeneous structures

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 199 - 223
Copyright
Copyright © The Association for Symbolic Logic 2019 

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