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, which is not necessarily commutative. If the Gromov–Hausdorff distance between two metric spaces, X and Y, is finite then their inverse semigroups M(X) and M(Y) (and hence their \(C^*\)-algebras) are isomorphic. We characterize the metrics that are idempotents, and give examples of metric spaces for which the semigroup M(X) (and the corresponding \(C^*\)-algebra) is commutative. 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 and the class of invertible metrics.

对于度量空间X，我们探讨的两个副本指标X。我们定义了此类度量的组成，并表明度量的等价类是半群M（X）。我们的主要结果是M（X）是逆半群。因此，可以定义此逆半群的\（C ^ * \）-代数，它不一定是可交换的。如果两个度量空间X和Y之间的Gromov–Hausdorff距离是有限的，则它们的逆半群M（X）和M（Y）（因此它们的\（C ^ * \）-代数）是同构的。我们表征了幂等度量，并给出了半群M（X）（以及对应的\（C ^ * \）-代数）可交换的度量空间的示例。我们还描述了由类的子集，确定指标的X中的希格森电晕子集的封锁方面X和类可逆指标。