Editing Noncrossing partition (section)

==Lattice structure==

Like the set of all [[Partition of a set|partitions of the set]] { 1, ..., ''n'' }, the set of all noncrossing partitions is a [[lattice (order)|lattice]] when [[partially ordered set|partially ordered]] by saying that a finer partition is "less than" a coarser partition.  However, although it is a subset of the lattice of all set partitions, it is ''not'' a sublattice, because the subset is not closed under the join operation in the larger lattice. In other words, the finest partition that is coarser than both of two noncrossing partitions is not always the finest ''noncrossing'' partition that is coarser than both of them.

Unlike the lattice of all partitions of the set, the lattice of all noncrossing partitions is self-dual, i.e., it is order-isomorphic to the lattice that results from inverting the partial order ("turning it upside-down").  This can be seen by observing that each noncrossing partition has a non-crossing complement. Indeed, every interval within this lattice is self-dual.