逆半群
Metrics on doubles as an inverse semigroup
摘要：
对于度量空间$X$，我们研究了$X$的两个副本上的度量。我们定义了这类度量的合成，并证明了度量的等价类是一个半群$M(X)$.我们的主要结果是$M(X)$是一个逆半群，因此，我们可以定义这个逆半群的$C^*$ -代数.我们刻画了幂等度量，找到了$M(X)$中的极小投影，并给出了度量空间的例子，使得半群$M(X)$是交换的.证明了如果两个度量空间$X$和$Y$之间的Gromov-Hausdorff距离是有限的，则$M(X)$和$M(Y)$是同构的.我们还描述了一类的度量确定的子集$X$的封闭的子集在希森电晕的$X$。
英文摘要：
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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