Like the set of all partitions of the set { 1, .., n }, the set of all noncrossing partitions is a lattice when partially ordered by saying that a finer partition is "less than" a coarser partition.
Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions of a set is self-dual, i.e., it is order-isomorphic to the lattice that results from inverting the partial order ("turning it upside-down").