Editing Graphic matroid (section)

==The lattice of flats==
The [[matroid|closure]] <math>\operatorname{cl}(S)</math> of a set <math>S</math> of edges in <math>M(G)</math> is a [[matroid|flat]] consisting of the edges that are not independent of <math>S</math> (that is, the edges whose endpoints are connected to each other by a path in <math>S</math>). This flat may be identified with the partition of the vertices of <math>G</math> into the [[Connected component (graph theory)|connected components]] of the subgraph formed by <math>S</math>: Every set of edges having the same closure as <math>S</math> gives rise to the same partition of the vertices, and <math>\operatorname{cl}(S)</math> may be recovered from the partition of the vertices, as it consists of the edges whose endpoints both belong to the same set in the partition. In the [[geometric lattice|lattice of flats]] of this matroid, there is an order relation <math>x\le y</math> whenever the partition corresponding to flat&nbsp;<math>x</math> is a refinement of the partition corresponding to flat&nbsp;<math>y</math>.

In this aspect of graphic matroids, the graphic matroid for a [[complete graph]] <math>K_n</math> is particularly important, because it allows every possible partition of the vertex set to be formed as the set of connected components of some subgraph. Thus, the lattice of flats of the graphic matroid of <math>K_n</math> is naturally isomorphic to the [[partition of a set|lattice of partitions of an <math>n</math>-element set]]. Since the lattices of flats of matroids are exactly the [[geometric lattice]]s, this implies that the lattice of partitions is also geometric.<ref>{{citation|title=Lattice Theory|volume=25|series=Colloquium Publications|publisher=American Mathematical Society|first=Garrett|last=Birkhoff|authorlink=Garrett Birkhoff|edition=3rd|year=1995|isbn=9780821810255|page=95|url=https://books.google.com/books?id=0Y8d-MdtVwkC&pg=PA95}}.</ref> <!-- test -->