论文标题
粗公制和统一度量
Coarse metric and uniform metric
论文作者
论文摘要
我们介绍了粗制度量的概念。每个粗制度量都会在基础集合上引起粗糙的结构。相反，我们观察到所有粗加度都以独特的方式均来自特定类型的粗制度量。 In the case when the coarse structure $\mathcal{E}$ on a set $X$ is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set $\mathcal{P}_0(X)$ of non-empty subsets of $X$ and show that it induces the Hausdorff coarse structure on $ \ MATHCAL {P} _0（x）$。 另一方面，我们定义了伪均匀度量的概念。每个伪均匀度量指标在基础空间上诱导均匀的结构。在相反的方向上，我们表明，$ x $上的均匀结构$ \ MATHCAL {u} $是由从$ x \ times x $到部分订购的集合的地图$ d $引起的（如果$ d $没有要求，并且仅在$ d $上没有要求），则仅当$ \ \ \ \ m nathcal {u} $ pass $ \ mathcal off base $ \ nath $ \ mathcal cup} \ {\ bigCap \ Mathcal {U} \} $在任意交叉点下关闭。在这种情况下，$ \ Mathcal {U} $实际上由伪统一度量定义。我们还表明，统一结构$ \ MATHCAL {u} $来自伪统一度量，该指标在且仅当$ \ Mathcal {u} $才能完全有序的情况下以完全有序的集合进行值。 最后，一个评估环将产生一个粗糙和伪均匀度量的示例，该指标将值以完全有序的集合为单位。
We introduce the notion of coarse metric. Every coarse metric induces a coarse structure on the underlying set. Conversely, we observe that all coarse spaces come from a particular type of coarse metric in a unique way. In the case when the coarse structure $\mathcal{E}$ on a set $X$ is defined by a coarse metric that takes values in a meet-complete totally ordered set, we define the associated Hausdorff coarse metric on the set $\mathcal{P}_0(X)$ of non-empty subsets of $X$ and show that it induces the Hausdorff coarse structure on $\mathcal{P}_0(X)$. On the other hand, we define the notion of pseudo uniform metric. Each pseudo uniform metric induces a uniform structure on the underlying space. In the reverse direction, we show that a uniform structure $\mathcal{U}$ on a set $X$ is induced by a map $d$ from $X\times X$ to a partially ordered set (with no requirement on $d$) if and only if $\mathcal{U}$ admits a base $\mathcal{B}$ such that $\mathcal{B}\cup \{\bigcap \mathcal{U}\}$ is closed under arbitrary intersections. In this case, $\mathcal{U}$ is actually defined by a pseudo uniform metric. We also show that a uniform structures $\mathcal{U}$ comes from a pseudo uniform metric that takes values in a totally ordered set if and only if $\mathcal{U}$ admits a totally ordered base. Finally, a valuation ring will produce an example of a coarse and pseudo uniform metric that take values in a totally ordered set.
