Editing Matroid (section)

===Hyperplanes===
In a matroid of rank <math> r</math>, a flat of rank <math>r - 1</math> is called a ''hyperplane''. (Hyperplanes are also called ''co-atoms'' or ''copoints''.) These are the maximal proper flats; that is, the only superset of a hyperplane that is also a flat is the set <math>E</math> of all the elements of the matroid. An equivalent definition is that a coatom is a subset of ''E'' that does not span ''M'', but such that adding any other element to it does make a spanning set.<ref name=w38-39>{{harvtxt|Welsh|1976|pp=38–39}}, Section&nbsp;2.2, "The Hyperplanes of a Matroid".</ref>

The family <math>\mathcal{H}</math> of hyperplanes of a matroid has the following properties, which may be taken as yet another axiomatization of matroids:<ref name=w38-39/>

* (H1) There do not exist distinct sets <math>X</math> and <math>Y</math> in <math>\mathcal{H}</math> with <math> X \subseteq Y</math>. That is, the hyperplanes form a [[Sperner family]].

* (H2) For every <math> x \in E </math> and distinct <math> Y, Z \in \mathcal{H} </math> with <math> x\notin Y \cup Z</math>, there exists <math> X \in \mathcal{H} </math> with <math> (Y \cap Z) \cup \{x\} \subseteq X</math>.