= Rank function of a partially ordered set
= poset rank
{synonym}
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>.
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