a particular instance of the previous example (number of boxes in the Young diagram), face lattices of convex polytopes (dimension of the face, plus one), abstract polytopes ("distance" from the least face, minus one), abstract simplicial complexes (number of elements of the simplex). Alternative characterizations A bounded poset admits a grading if and only if all maximal chains in P have the same length: setting the rank of the least element to 0 then determines the rank function completely. This covers many finite cases of interest; see picture for a negative example. However, unbounded posets can be more complicated. A candidate rank function, compatible with the ordering, makes a poset into graded poset if and only if, whenever one has x < z with z of rank n + 1, an element y of rank n can be found with x ≤ y < z. This condition is sufficient because if z is taken to be a cover of x, the only possible choice is y = x showing that the ranks of x and z differ by 1, and it is necessary because in a graded poset one can take for y any element of maximal rank with x ≤ y < z, which always exists and is covered by z. Often a poset comes with a natural candidate for a rank function; for instance if its elements are finite subsets of some base set B, one can take the number of elements of those subsets. Then the criterion just given can be more practical than the definition because it avoids mention of covers. For instance if B is itself a poset, and P consists of its finite lower sets (subsets for which with every one of its elements, all smaller elements are also in the subset), then the criterion is automatically satisfied, since for lower sets x ⊆ z there is always a maximal element of z that is absent from x, and it can be removed from z to form y. In some common posets such as the face lattice of a convex polytope there is a natural grading by dimension, which if used as rank function would give the minimal element, the empty face, rank −1. In such cases it might be convenient to bend the definition stated above by adjoining the value −1 to the set of values allowed for the rank function. Allowing arbitrary