Leigh.Samphier/Sandbox/Equivalence of Definitions of Matroid Circuit Axioms

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Theorem

Let $S$ be a finite set.

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


The following are equivalent:

Condition 1

$\mathscr C$ satisfies the circuit axioms:

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


Condition 2

$\mathscr C$ satisfies the circuit axioms:

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


Condition 3

$\mathscr C$ satisfies the circuit axioms:

\((C1)\)   $:$   \(\displaystyle \O \notin \mathscr C \)             
\((C2)\)   $:$     \(\displaystyle \forall C_1, C_2 \in \mathscr C:\) \(\displaystyle C_1 \neq C_2 \implies C_1 \not \subseteq C_2 \)             
\((C3'')\)   $:$     \(\displaystyle \forall X \subseteq S \land \forall x \in S:\) \(\displaystyle \paren{\forall C \in \mathscr C : C \not \subseteq X} \implies \paren{\exists \text{ at most one } C \in \mathscr C : C \subseteq X \cup \set x} \)             


Condition 4

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


Proof

Condition 1 implies Condition 2

$\Box$

Condition 2 implies Condition 3

$\Box$

Condition 3 implies Condition 4

$\Box$

Condition 4 implies Condition 1

$\blacksquare$

Sources