There is a relationship between rank-based statistics and the notion of an empirical distribution function. The empirical cumulative distribution function can be written as
\[\hat F (t) \triangleq \frac{1}{n} \sum_{i=1}^n \mathbb{I}_{X_i \leq t}\]
Note that \[\text{rank}(x) = \sum_{i=1}^n \mathbb{I}_{X_i \leq x}\] is a rank, and thus we have
\[\hat F (x) = \frac{\text{rank}(x)}{n}\]
where we have restricted the parameter \(t\) to the subset of the domain of \(\hat F\) that are observed.
We can have cumulative distribution functions on things that are partially-ordered, not just subsets of the real numbers.
The real numbers are totally-ordered, but note that every total order is a partial order.
The empirical cumulative distribution function according to GCT uniformly converges to the true cumulative distribution \(F\) if it exists.