In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties: The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1. The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value. Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram. Examples Some examples of graded posets (with the rank function in parentheses) are: the natural numbers N with their usual order (rank: the number itself), or some interval [0, N] of this poset, Nn, with the product order (sum of the components), or a subposet of it that is a product of intervals, the positive integers, ordered by divisibility (number of prime factors, counted with multiplicity), or a subposet of it formed by the divisors of a fixed N, the Boolean lattice of finite subsets of a set (number of elements of the subset), the lattice of partitions of a set into finitely many parts, ordered by reverse refinement (number of parts), the lattice of partitions of a finite set X, ordered by refinement (number of elements of X minus number of parts), a group and a generating set, or equivalently its Cayley graph, ordered by the weak or strong Bruhat order, and ranked by word length (length of shortest reduced word). In particular for Coxeter groups, for example permutations of a totally ordered n-element set, with either the weak or strong Bruhat order (number of adjacent inversions), geometric lattices, such as the lattice of subspaces of a vector space (dimension of the subspace), the distributive lattice of finite lower sets of another poset (number of elements), the poset of all unlabeled posets on (number of elements), Young's lattice,