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Preprint Number 926
926. Gabriel Conant
Extending partial isometries of generalized metric spaces
Submission date: 16 September 2015.
We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class K of finite generalized metric spaces satisfies the Hrushovski extension property:
for any A in K there is some B in K such that A is a subspace of B and any partial isometry of A extends to a total isometry of B. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid R. When R is also countable, this can be used to show that the isometry group of the Urysohn space over R has ample generics.
Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd n ≥ 3, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by n.
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