PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Extending partial isometries of generalized metric spaces

Volume 244 / 2019

Gabriel Conant Fundamenta Mathematicae 244 (2019), 1-16 MSC: Primary 03C13; Secondary 05C12, 05C38. DOI: 10.4064/fm484-9-2018 Published online: 26 October 2018

Abstract

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal {K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in \mathcal {K}$ there is some $B\in \mathcal {K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. We prove the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal {R}$. When $\mathcal {R}$ is also countable, we use this to show that the isometry group of the Urysohn space over $\mathcal {R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting metric triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.

Authors

  • Gabriel ConantDepartment of Mathematics
    University of Notre Dame
    Notre Dame, IN 46556, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image