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 (with positive integer ranks) 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 (for positive integer rank functions) compatibility of ρ with the ordering follows from the requirement about covers. As a variant of the definition of a graded poset, Birkhoff allows rank functions to have arbitrary (rather than only nonnegative) integer values. In this variant, the integers can be graded (by the identity function) in his setting, and the compatibility of ranks with the ordering is not redundant. As a third variant, Brightwell and West define a rank function to be integer-valued, but don't require its compatibility with the ordering; hence this variant can grade even e.g. the real numbers by any function, as the requirement about covers is vacuous for this example. 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 [Ô, x] have the same length. This condition is necessary since every step in a maximal chain is a covering relation, which should change the rank