A graded poset is a poset $P$ that is equipped with a rank function $\rho$, which is a function from $P$ to $\mathbb{Z}$, satisfying the following three conditions:

1. 1.

$\rho$ is constant on all minimal elements of $P$, usually with value $-1$ or $0$

2. 2.

$\rho$ is isotone, that is, if $a\leq b$, then $\rho(a)\leq\rho(b)$, and

3. 3.

$\rho$ preserves covering relations: if $a\prec b$, then $\rho(a)+1=\rho(b)$.

Equivalently, a poset $P$ is graded if it admits a partition into maximal antichains $\{A_{n}\mid n\in\mathbb{N}\}$ such that for each $x\in A_{n}$, all of the elements covering $x$ are in $A_{n+1}$ and all the elements covered by $x$ are in $A_{n-1}$.

A poset $P$ can be graded if one can define a rank function $\rho$ on $P$ so $(P,\rho)$ is a graded poset. Below is a poset that can not be graded:

 $\entrymodifiers={[o]}\xymatrix@!=1pt{&&\circ\ar@{-}[ld]\ar@{-}[rd]&\\ &\ar@{-}[ld]&&\circ\ar@{-}[d]\\ \circ\ar@{-}[rd]&&&\ar@{-}[d]\\ &\ar@{-}[rd]&&\circ\ar@{-}[ld]\\ &&\circ&}$

## Generalized rank functions

Since certain common posets such as the face lattice of a polytope are most naturally graded by dimension (http://planetmath.org/Dimension2), the rank of a minimal element is sometimes required to be $-1$.

More generally, given a chain $C$, one can define $C$-graded posets. A poset $P$ is $C$-graded provided that there is a poset map $\rho\colon P\to C$ that preserves covers and is constant on minimal elements of $P$. Such a rank function is unique up to choice of the rank of minimal elements. In practice, however, the term graded is only used to indicate $\mathbb{N}$-grading, $\mathbb{N}\cup\{-1\}$-grading, or $\mathbb{Z}$-grading.

## Maximal chains in graded posets

Let $P$ be a graded poset with rank function $\rho$. A chain $C$ in $P$ is said to be a saturated chain provided that $\rho(C)=\rho(P)$. If $C$ is saturated in $P$, then each cover relation in $C$ is also a cover relation in $P$; thus a saturated chain is also a maximal chain.

It is a property of graded posets that all saturated chains have the same cardinality. As a partial converse, if $P$ is a finite bounded poset (http://planetmath.org/BoundedLattice) and each maximal chain has the same cardinality, then $P$ is graded.

 Title graded poset Canonical name GradedPoset Date of creation 2013-03-22 14:09:12 Last modified on 2013-03-22 14:09:12 Owner mps (409) Last modified by mps (409) Numerical id 9 Author mps (409) Entry type Definition Classification msc 06A06 Classification msc 05B35 Related topic EulerianPoset Related topic StarProduct Related topic HeightOfAnElementInAPoset Defines rank function Defines saturated chain