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 $M$ if and only if:

$\forall x \in S \setminus A : \map \rho {A \cup \set x} = \map \rho A + 1$

Also known as

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

Sources

• 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid