Editing Partition of a set (section)

==Noncrossing partitions==
A partition of the set ''N'' = {1, 2, ..., ''n''} with corresponding equivalence relation ~ is '''[[noncrossing partition|noncrossing]]''' if it has the following property: If four elements ''a'', ''b'', ''c'' and ''d'' of ''N'' having ''a'' < ''b'' < ''c'' < ''d'' satisfy ''a'' ~ ''c'' and ''b'' ~ ''d'', then ''a'' ~ ''b'' ~ ''c'' ~ ''d''. The name comes from the following equivalent definition: Imagine the elements 1, 2, ..., ''n'' of ''N'' drawn as the ''n'' vertices of a regular ''n''-gon (in counterclockwise order). A partition can then be visualized by drawing each block as a polygon (whose vertices are the elements of the block). The partition is then noncrossing if and only if these polygons do not intersect.

The lattice of noncrossing partitions of a finite set forms a subset of the lattice of all partitions, but not a sublattice, since the join operations of the two lattices do not agree.

The noncrossing partition lattice has taken on importance because of its role in [[free probability]] theory.