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.Not containing arbitrarily long descending chains starting at a fixed element of course excludes any infinite descending chains. The former condition is strictly stronger though; the set \N\cup\{\infty\} has arbitrarily long chains descending from \infty, but has no 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'Lattice Theory', Am. Math. Soc., Colloquium Publications, Vol.25, 1967, p.5 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 WestSee reference [2], p.722. 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.
