Editing Matroid (section)

=== Flats<span class="anchor" id="closed_sets_flats_anchor"></span> ===
A set whose closure equals itself is said to be ''closed'', or a ''flat'' or ''subspace'' of the matroid.<ref>{{harvtxt|Welsh|1976|pp=21–22}}, Section&nbsp;1.8, "Closed sets = Flats = Subspaces".</ref> A set is closed if it is [[maximal element|maximal]] for its rank, meaning that the addition of any other element to the set would increase the rank. The closed sets of a matroid are characterized by a covering partition property:

* (F1) The whole point set <math>E</math> is closed.

* (F2) If <math>S</math> and <math>T</math> are flats, then <math>S\cap T</math> is a flat.

* (F3) If <math>S</math> is a flat, then each element of <math>E\smallsetminus S</math> is in precisely one of the flats <math>T</math> that [[Covering relation|cover]] <math>S</math> (meaning that <math>T</math> properly contains <math>S</math>, but there is no flat <math>U</math> between <math>S</math> and <math>T</math>).

The class <math>\mathcal{L}(M)</math> of all flats, [[partially ordered set|partially ordered]] by set inclusion, forms a [[matroid lattice]]. Conversely, every matroid lattice <math>L</math> forms a matroid over its set <math>E</math> of [[Atom (order theory)|atoms]] under the following closure operator:  for a set <math>S</math> of atoms with join <math> \bigvee S</math>, 
:<math>\operatorname{cl}(S) = \{ x\in E\mid x\le\bigvee S \}</math>. 
The flats of this matroid correspond one-for-one with the elements of the lattice; the flat corresponding to lattice element <math>y</math> is the set
:<math>\{ x\in E\mid x\le y\}</math>. 
Thus, the lattice of flats of this matroid is naturally isomorphic to <math>L</math>.