Graded Poset - Alternative Characterizations

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. 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 integers as rank would however give a fundamentally different notion; for instance the existence of a minimal element would no longer be assured.

A graded poset cannot have any elements x for which arbitrarily long chains with greatest element x exist, as otherwise it would have to have elements of arbitrarily small (and eventually negative) rank. For instance, the integers (with the usual order) cannot be a graded poset, nor can any interval (with more than one element) of rational or real numbers. (In particular, graded posets are well-founded, meaning that they satisfy the descending chain condition (DCC): they do not contain any infinite descending chains.) Henceforth we shall therefore only consider posets in which this does not happen. This implies that whenever x < y we can get from x to y by repeatedly choosing a cover, finitely many times. It also means that compatibility of ρ with the ordering follows from the requirement about covers.

Note that graded posets need not satisfy the ascending chain condition (ACC): for instance, the natural numbers contain the infinite ascending chain .

A poset is graded if and only if every connected component of its comparability graph is graded, so further characterizations will suppose this comparability graph to be connected. On each connected component the rank function is only unique up to a uniform shift (so the rank function can always be chosen so that the elements of minimal rank in their connected component have rank 0).

If P has a least element Ô then being graded is equivalent to the condition that for any element x all maximal chains in the interval have the same length. This condition is necessary since every step in a maximal chain is a covering relation, which should change the rank by 1. The condition is also sufficient, since when it holds, one can use the mentioned length to define the rank of x (the length of a finite chain is its number of "steps", so one less than its number of elements), and whenever x covers y, adjoining x to a maximal chain in gives a maximal chain in .

If P also has a greatest element Î (so that it is a bounded poset), then the previous condition can be simplified to the requirement that all maximal chains in P have the same (finite) length. This suffices, since any pair of maximal chains in can be extended by a maximal chain in to give a pair of maximal chains in P.

Note Stanley defines a poset to be graded of length n if all its maximal chains have length n (Stanley 1997, p.99). This definition is given in a context where interest is mostly in finite posets, and although the book subsequently often drops the part "of length n", it does not seem appropriate to use this as definition of "graded" for general posets, because (1) it says nothing about posets whose maximal chains are infinite, in particular (2) it excludes important posets like Young's lattice. Also it is not clear why in a graded poset all minimal elements, as well as all maximal elements, should be required to have the same length, even if Stanley gives examples making clear that he does mean to require that (ibid, pp.216 and 219).