Extending partial isometries of generalized metric spaces

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 $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used 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 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 $n$.