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, 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.