Leigh.Samphier/Sandbox/Definition:Circuit Axioms (Matroid)/Definition 1

Definition

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

$\mathscr C$ is said to satisfy the circuit axioms if and only if:

$(C1)$	$:$		$\displaystyle \O \notin \mathscr C $
$(C2)$	$:$	$\displaystyle \forall C_1, C_2 \in \mathscr C:$	$\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 $
$(C3)$	$:$	$\displaystyle \forall C_1, C_2 \in \mathscr C:$	$\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z $

1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid, Theorem 5

\((C1)\)	$:$		\(\displaystyle \O \notin \mathscr C \)
\((C2)\)	$:$	\(\displaystyle \forall C_1, C_2 \in \mathscr C:\)	\(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)
\((C3)\)	$:$	\(\displaystyle \forall C_1, C_2 \in \mathscr C:\)	\(\displaystyle C_1 \neq C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren{C_1 \cup C_2} \setminus \set z \)