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A 'graded poset', sometimes called a 'ranked poset' (but see the article for an alternative meaning), is a partially ordered set (poset) ''P'' in which every maximal chain is finite and has the same length. This property is called the 'chain condition'.
Every graded poset has an integer-valued 'rank function' $$
ho:P omathbb{Z} which satisfies the following properties:

★ If ''x'' and ''y'' are both minimal elements, then $$
ho(x)=
ho(y)=0.

★ If ''y'' covers ''x'', then $$
ho(y)=
ho(x)+1.
Equivalently, a poset ''P'' is graded if it admits a partition into maximal antichains $A\_n$ for ''n'' = 0, 1, ..., ''r'' (where ''r'' is a nonnegative integer) such that for each $xin\; A\_n$, all of the elements covering ''x'' are in $A\_\{n+1\}$ and all the elements covered by ''x'' are in $A\_\{n-1\}$. The 'rank of' ''P'', written $$
ho(P), is the maximum rank of any element. The rank of a minimal element is 0.
In some common posets such as the face lattice of a convex polytope there is a natural grading by dimension, which is 1 less than the rank; thus, in the face lattice, the rank of the empty face is 0 but its dimension is -1. Sometimes it is most convenient to use dimension rather than rank in the grading.

Contents

Characteristic polynomial

Examples

Characteristic polynomial

A finite graded poset ''P'' with a unique minimal element (called 0) has an important invariant called its 'characteristic polynomial'. Let μ be the Möbius function of ''P''. The characteristic polynomial is
:$p\_P(lambda)\; :=\; sum\_\{x\; in\; P\}\; mu(0,x)\; lambda^\{$
ho(L)-
ho(x)} .
The characteristic polynomial of a matroid is the characteristic polynomial of its lattice of flats, except that if the empty set is not closed, then the characteristic polynomial is 0.

Examples

Some examples of graded posets are:

★ geometric lattices of finite rank, such as the lattice of subspaces of a vector space of finite dimension,

★ finite distributive lattices,

★ face lattices of convex polytopes,

★ face posets of finite-dimensional cell complexes,

★ pure abstract simplicial complexes.