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

2.
$\rho $ is isotone, that is, if $a\le b$, then $\rho (a)\le \rho (b)$, and

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 ${A}_{n1}$.
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:
$$\text{entrymodifiers}=[o]\text{xymatrix}\mathrm{@}!=1pt\mathrm{\&}\mathrm{\&}\circ \text{ar}\mathrm{@}[ld]\text{ar}\mathrm{@}[rd]\mathrm{\&}\mathrm{\&}\text{ar}\mathrm{@}[ld]\mathrm{\&}\mathrm{\&}\circ \text{ar}\mathrm{@}[d]\circ \text{ar}\mathrm{@}[rd]\mathrm{\&}\mathrm{\&}\mathrm{\&}\text{ar}\mathrm{@}[d]\mathrm{\&}\text{ar}\mathrm{@}[rd]\mathrm{\&}\mathrm{\&}\circ \text{ar}\mathrm{@}[ld]\mathrm{\&}\mathrm{\&}\circ \mathrm{\&}$$ 
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 $\rho :P\to C$ 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  20130322 14:09:12 
Last modified on  20130322 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 