Definition:Flat (Matroid)

Definition
Let $M = \struct{S, \mathscr I}$ be a matroid.

Let $\rho : \powerset S \to \Z$ be the rank function of $M$. A subset $A \subseteq S$ is a flat of $M$ :
 * $\forall x \in S \setminus A : \map \rho {A \cup \set x} = \map \rho A + 1$

Also known as
In some sources a flat of $M$ is called a closed set or a subspace of the matroid $M$.