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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.



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

Also defines:  rank function, saturated chain
Keywords:  poset, graded, rank
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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 4999 times total.

Classification:
AMS MSC: 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)
 05B35 (Combinatorics :: Designs and configurations :: Matroids, geometric lattices)
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