Graded Poset - Examples

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 of this poset,
• Nn, with the product order (sum of the coefficients), 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),
• Young's lattice, a particular instance of the previous example (number of boxes in the Young diagram),
• face lattices of convex polytopes (dimension of the face, plus one),
• abstract polytopes ("distance" from the least face, minus one ),
• abstract simplicial complexes (number of elements of the simplex).