pure poset

A poset is pure if it is finite and every maximal chain has the same length.If P is a pure poset, we can create a rank function r on P by definingr(x) to be the length of a maximal chain bounded above by x. Everyinterval of a pure poset is a graded poset, and every graded poset is pure. Moreover,the closure of a pure poset, formed by adjoining an artificial minimum element andan artificial maximum element, is always graded.

The face poset of a pure simplicial complex is pure as a poset.