Concept
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. 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).
