Ranked partially ordered set

A ranked partially ordered set has an integer-valued rank function of a partially ordered set that increases by one whenever $y$ covers $x$, meaning $x<y$ with no element strictly between them. For finite orders with minimal elements, one commonly normalizes every minimal element to rank zero. Conventions about requiring all maximal chains to have the same length should be stated separately.

In the Boolean lattice, $\rho (S)=|S|$: a covering step adds one element. In the positive divisors of $12$ ordered by divisibility, $\rho (d)$ is the number of prime factors counted with multiplicity. Rank levels organize an order into layers and allow questions about its Sperner property.

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.