View source for Rank of a partially ordered set

The maximum cardinality of a [[chain]] in a [[partially ordered set]].  If $(P,\le)$ is a partially ordered set then the rank of $b$ relative to $a$, where $a \le b$, is the rank of the [[interval]] $[a,b]$; if $P$ has a minimum element $0$, then the rank $\rho(x)$ of an element $x$ is the rank of $x$ relative to $0$.  A chain $C$ in $P$ is maximal if no proper superset of $C$ is a chain.  If all maximal chains have the same cardinality, then $P$ is said to be ''graded'' by rank.

If the rank of each element of the poset $(P,\le)$ is finite, and there is a finite number $p_n$ of elements of rank $n$ (in particular, if $P$ is finite) then the rank generating function of $P$ is the formal series
$$
\sum_{n=0}^\infty p_n z^n \ .
$$

A [[lattice]] with a rank function $\rho$ is ''(upper)'' ''[[Semi-modular lattice|semi-modular]]'' if 
$$
\rho(x) + \rho(y) \ge \rho(x \vee y) + \rho(x \wedge y)
$$
and ''[[Modular lattice|modular]]'' if
$$
\rho(x) + \rho(y) = \rho(x \vee y) + \rho(x \wedge y) \ .
$$


====References====
* Stanley, Richard P. ''Enumerative combinatorics'' '''I''' Wadsworth & Brooks/Cole (1986) {{ISBN|0-534-06546-5}} {{ZBL|0608.05001}}

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