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'graded poset'
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| Title of object: |
graded poset |
| Canonical Name: |
GradedPoset |
| Type: |
Definition |
| Created on: |
2004-02-12 20:39:14 |
| Modified on: |
2006-12-05 00:32:48 |
| Classification: |
msc:06A06, msc:05B35 |
| Keywords: |
poset, graded, rank |
| Defines: |
rank function, saturated chain |
Preamble:
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Content:
\PMlinkescapeword{satisfy}
\PMlinkescapeword{properties}
\PMlinkescapeword{rank}
\PMlinkescapeword{theory}
\PMlinkescapeword{saturated}
\PMlinkescapeword{property}
A \emph{graded poset} is a poset $P$ that is equipped with a \emph{rank function} $\rho\colon P\to\mathbb{Z}$. The rank function must send all minimal elements of $P$ to the same element of $\mathbb{Z}$ (usually $-1$ or $0$) and must also preserve covering relations.
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 $A_{n-1}$.
\subsection*{Generalized rank functions}
Since certain common posets such as the face lattice of a polytope are most naturally graded by \PMlinkname{dimension}{Dimension2}, the rank of a minimal element is sometimes required to be $-1$.
More generally, given a chain $C$, one can define \emph{$C$-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.
\subsection*{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 \emph{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 \PMlinkname{bounded poset}{BoundedLattice} and each maximal chain has the same cardinality, then $P$ is graded. |
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