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ON EXTENSIONS OF PARTIAL ISOMORPHISMS

Part of: Model theory Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  20 July 2020

MAHMOOD ETEDADIALIABADI  and
SU GAO
MAHMOOD ETEDADIALIABADI
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH TEXAS 1155 UNION CIRCLE #311430, DENTON, TX76203, USAE-mail: mahmood.etedadialiabadi@unt.eduE-mail: sgao@unt.edu
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Abstract

In this paper we study a notion of HL-extension (HL standing for Herwig–Lascar) for a structure in a finite relational language $\mathcal {L}$ . We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$ -structure has a dense locally finite subgroup.

Keywords

Hrushovski propertyextension property for partial automorphisms (EPPA)partial isomorphismHL-extensionHL-mapcoherentultraextensiveultrahomogeneouslocally finiteHenson graph

MSC classification

Primary: 03C13: Finite structures 03C55: Set-theoretic model theory
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E26: Residual properties and generalizations; residually finite groups
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 1 , March 2022 , pp. 416 - 435
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Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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