Coarse metric and uniform metric

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.
Comment: This version contains the following minor corrections to the published version: (1). The word "increasing" is added in Definition 2(c). (2). In part (a) of Theorem 5, as well as parts (c) and (e) of Proposition 8, we need to assume that the coarse metric take values in a meet complete directed set. The corrections and changes are marked in red