Can something have more than one ranking in a partial order?

Question

Suppose $(P,\leq)$ is a partial order. How does one rigorously define a ranking in a partial order? So, for instance, the top element would have a rank of 1. It could also be that some elements' ranks are undefined. Could two or more elements have the same rank? Also, can some element be assigned for more than one rank? For example, an element might have a rank of both 4th and 5th. Is there literature on generalizing a notion of ranking, from linear orders to partial orders, preorders, and perhaps even arbitrary relations?