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 $A_{n-1}$.

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:

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	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