PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
graded poset (Definition)

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:

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

Generalized rank functions

Since certain common posets such as the face lattice of a polytope are most naturally graded by dimension, 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 and each maximal chain has the same cardinality, then $ P$ is graded.



Anyone with an account can edit this entry. Please help improve it!

"graded poset" is owned by mps. [ full author list (2) ]
(view preamble)

View style:

See Also: Eulerian poset, star product, height of an element in a poset

Also defines:  rank function, saturated chain
Keywords:  poset, graded, rank
Log in to rate this entry.
(view current ratings)

Cross-references: finite, converse, cardinality, term, covers, map, chain, polytope, lattice, face, maximal antichains, partition, relations, covering, preserves, minimal elements, function, poset
There are 10 references to this entry.

This is version 6 of graded poset, born on 2004-02-12, modified 2007-01-03.
Object id is 5571, canonical name is GradedPoset.
Accessed 4677 times total.

Classification:
AMS MSC: 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)
 05B35 (Combinatorics :: Designs and configurations :: Matroids, geometric lattices)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)