Axiom:Base Axiom (Matroid)

Definition

Let $S$ be a finite set.

Let $\mathscr B$ be a non-empty set of subsets of $S$.

$\mathscr B$ is said to satisfy the base axiom if and only if:

\((\text B 1)\)

$:$

\(\ds \forall B_1, B_2 \in \mathscr B:\)

\(\ds x \in B_1 \setminus B_2 \implies \exists y \in B_2 \setminus B_1 : \paren {B_1 \cup \set y} \setminus \set x \in \mathscr B \)

Sources

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