Category:Equivalence of Definitions of Matroid Circuit Axioms

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

The following statements are equivalent:

$\mathscr C$ satisfies the circuit axioms:

$(\text C 1)$	$:$		$\ds \O \notin \mathscr C $
$(\text C 2)$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 $
$(\text C 3)$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\ds C_1 \ne 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 $

$\mathscr C$ satisfies the circuit axioms:

$(\text C 1)$	$:$		$\ds \O \notin \mathscr C $
$(\text C 2)$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 $
$(\text C 3')$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z $

$\mathscr C$ satisfies the circuit axioms:

$(\text C 1)$	$:$		$\ds \O \notin \mathscr C $
$(\text C 2)$	$:$	$\ds \forall C_1, C_2 \in \mathscr C:$	$\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 $
$(\text C 3)$	$:$	$\ds \forall X \subseteq S \land \forall x \in S:$	$\ds \paren {\forall C \in \mathscr C : C \nsubseteq X} \implies \paren {\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} $

$\mathscr C$ is the set of circuits of a matroid on $S$

Pages in category "Equivalence of Definitions of Matroid Circuit Axioms"

The following 16 pages are in this category, out of 16 total.

\((\text C 1)\)	$:$		\(\ds \O \notin \mathscr C \)
\((\text C 2)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)
\((\text C 3)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \implies \exists C_3 \in \mathscr C : C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)

\((\text C 1)\)	$:$		\(\ds \O \notin \mathscr C \)
\((\text C 2)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)
\((\text C 3')\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \land z \in C_1 \cap C_2 \land w \in C_1 \setminus C_2 \implies \exists C_3 \in \mathscr C : w \in C_3 \subseteq \paren {C_1 \cup C_2} \setminus \set z \)

\((\text C 1)\)	$:$		\(\ds \O \notin \mathscr C \)
\((\text C 2)\)	$:$	\(\ds \forall C_1, C_2 \in \mathscr C:\)	\(\ds C_1 \ne C_2 \implies C_1 \nsubseteq C_2 \)
\((\text C 3)\)	$:$	\(\ds \forall X \subseteq S \land \forall x \in S:\)	\(\ds \paren {\forall C \in \mathscr C : C \nsubseteq X} \implies \paren {\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \)