Metrics on doubles as an inverse semigroup

For a metric space $X$ we study metrics on the two copies of $X$. We define composition of such metrics and show that the equivalence classes of metrics are a semigroup $M(X)$ Our main result is that $M(X)$ is an inverse semigroup, therefore, one can define the $C^*$-algebra of this inverse semigroup. We characterize the metrics that are idempotents, find a minimal projection in $M(X)$ and give examples of metric spaces, for which the semigroup $M(X)$ is commutative. We show that if the Gromov-Hausdorff distance between two metric spaces, $X$ and $Y$, is finite then $M(X)$ and $M(Y)$ are isomorphic. We also describe the class of metrics determined by subsets of $X$ in terms of the closures of the subsets in the Higson corona of $X$.

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