Graded poset

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A locally finite graded poset consists of a poset P along with a function \rho:P\to\mathbb{Z}, called the rank function of P. The rank function must satisfy the following properties:

  • If x and y are both minimal elements, then ρ(x) = ρ(y).
  • If y covers x, then ρ(y) = ρ(x) + 1.

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 An + 1 and all the elements covered by x are in An − 1.

The rank function is unique up to choice of the rank of minimal elements. In general poset theory, the rank of a minimal element is 0. However, since certain common posets such as the face lattice of a polytope are most naturally graded by dimension, sometimes the rank of a minimal element is required to be -1.

This article incorporates material from GradedPoset on PlanetMath, which is licensed under the GFDL.

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