 Rank function of a partially ordered set

A rank function of a partially ordered set assigns an integer $\rho (x)$ to each element so that $\rho (y)=\rho (x)+1$ whenever $y$ covers $x$. In a finite ranked partially ordered set with minimal elements normalized to zero, $\rho (x)$ is the length of every saturated chain from a minimal element to $x$.

For subsets ordered by inclusion, cardinality is a rank function of a partially ordered set. For a chain $a<b<c$, the normalized ranks are $0,1,2$. An order with two saturated chains of unequal lengths between the same endpoints cannot have a rank function of a partially ordered set, because the required increments would assign conflicting values. This notion is distinct from the rank of a set in cumulative set theory.