Metrics on doubles as an inverse semigroup

TL;DR

This paper introduces a novel inverse semigroup structure on metrics defined on two copies of a metric space, exploring algebraic properties, characterizations, and implications for metric space comparisons.

Contribution

It establishes that the set of equivalence classes of such metrics forms an inverse semigroup and explores its algebraic and topological properties, including isomorphisms and characterizations.

Findings

$M(X)$ is an inverse semigroup for any metric space $X$

Characterization of idempotent metrics and minimal projections in $M(X)$

Isomorphism of $M(X)$ and $M(Y)$ when Gromov-Hausdorff distance is finite

Abstract

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 $X$.