What is a rank? I've been thinking about unifying some notions related to ranking, order theory, ordinal data, and graded posets. While the notion of a grade in order theory is quite general, in some sense the way we talk about ranks is more general. I would like to bring rank-based statistics and order theory together in my theory development and analysis of data.
I would really like to know if there are others (especially but not exclusively in the literature) that have attempted to define ranks in a way that generalizes or interoperates with concepts in order theory.

I've included my attempt at defining a rank in order to illustrate some of the properties I'm interested in, but this should not be taken as a definitive answer. This definition reflects my cobbled together thinking from examples I have seen of "ranks".

Assume a collection random variables $\{X_1(\omega), \ldots, X_n(\omega) \}$ on outcome space $\Omega$, and partial order $\leq$. An  abstract ranking $\rho: \prod_{i=1}^n X_i(\omega) \mapsto \mathbb{R}_{\geq 0}^n$ is a function such that there exists a non-decreasing function $\kappa:\mathbb{N} \mapsto \mathbb{R}_{\geq0}$ that satisfies $\rho(\vec x)_i \leq \kappa(n)$ for all $i\in \{1, \ldots, n\}$. It must also hold that $\rho(\vec x)_i \leq \rho(\vec x)_j \iff x_i \leq x_j$ for all $i,j \in \{1, \ldots, n\}$ and for all $\omega \in \Omega$ exor $\rho(\vec x)_i \geq \rho(\vec x)_j \iff x_i \leq x_j$ for all $i,j \in \{1, \ldots, n\}$ and for all $\omega \in \Omega$. An component of an image element of an abstract ranking is called an abstract rank.

Here are some rationalizations for why I tried to define ranks this way.

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*Why is $\rho$ non-negative? I think for three reasons:

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*It is (for me) a little easier to keep track of positive numbers.

*Empirical induction: all the examples I've seen of "ranks" or "grade" meet this criterion.

*Allowing $\rho$ to have zero in its image can be a programing convenience where the rank can coincide with the indices of a data structure of sorted elements. Is adding one so difficult? No, but zero doesn't bother me either.

*Aesthetic: This feels right. ¯_(ツ)_/¯



*Why monotonic rather than nondecreasing in particular? While grades are order-preserving, sometimes we rank quantities in an order-reversing fashion. For example, in a contest of weight lifting the 1st place might go to the person who lifts the most weight.

*Why this bounding $\kappa$ function? A function being monotone and non-negative just didn't seem specific enough. I've noted as a matter of empirical induction that such a bound occurs with what have been usually described as ranks, and it is likewise true of grades, so I am content to include it.


I'm still ruminating about this potentially additional property:

It is also required of $\kappa$ that for any finite $n$ there exists $\omega$ in outcome space $\Omega$ such that $\max_i \rho (\vec x_{\omega})_i = \kappa(n)$.

