Recently I came across a paper by Stephen McAdam "GRADE SCHEMES AND GRADE FUNCTIONS" (Transactions of the American Mathematical Society, Vol. 288, No. 2 (Apr., 1985), pp. 563-590). So far I have just skimmed it, but from what I've understood he tries to set up unified axiomatic approach to study codimension, depth, asymptotic grade, etc. All these functions can be considered as monotone functions from the poset of prime ideals $P$ of the ring into the natural numbers satisfying certain properties (see Lemma 1.2 and Theorem 2.4 in the paper).

I wonder is it possible to recover (if not fully, then to some extent) the poset $P$ from knowledge of such functions? i.e. does there exist a theorem (in poset theory?) which states that study of $P$ is equivalent to study of "grade functions" whose abstract characterization is given in the paper.

More interestingly, I would like to know what is the meaning of such functions in terms of category of $R$-modules. There are some answers in this direction, see for example http://arxiv.org/abs/1202.5605 (CLASSIFICATION OF RESOLVING SUBCATEGORIES AND GRADE CONSISTENT FUNCTIONS HAILONG DAO AND RYO TAKAHASHI). I have not read this paper either.

I have asked a similar question here: Characterizing Posets by Functions Into Natural Numbers

Thanks!