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ON EXTENSIONS OF PARTIAL ISOMORPHISMS
Published online by Cambridge University Press: 20 July 2020
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.
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- © The Author(s), 2020. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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