Axiom:Rank Axioms (Matroid)

Definition

Let $\rho : \powerset S \to \Z$ be a mapping from the power set of $S$ into the integers.

$\rho$ is said to satisfy the rank axioms if and only if:

$(\text R 1)$	$:$		$\ds \map \rho \O = 0 $
$(\text R 2)$	$:$	$\ds \forall X \in \powerset S \land y \in S:$	$\ds \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1 $
$(\text R 3)$	$:$	$\ds \forall X \in \powerset S \land y, z \in S:$	$\ds \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set y \cup \set z} = \map \rho X $

$\rho$ is said to satisfy the rank axioms if and only if:

$(\text R 4)$	$:$	$\ds \forall X \in \powerset S:$	$\ds 0 \le \map \rho X \le \size X $
$(\text R 5)$	$:$	$\ds \forall X, Y \in \powerset S:$	$\ds X \subseteq Y \implies \map \rho X \le \map \rho Y $
$(\text R 6)$	$:$	$\ds \forall X, Y \in \powerset S:$	$\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y $

\((\text R 1)\)	$:$		\(\ds \map \rho \O = 0 \)
\((\text R 2)\)	$:$	\(\ds \forall X \in \powerset S \land y \in S:\)	\(\ds \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1 \)
\((\text R 3)\)	$:$	\(\ds \forall X \in \powerset S \land y, z \in S:\)	\(\ds \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set y \cup \set z} = \map \rho X \)

\((\text R 4)\)	$:$	\(\ds \forall X \in \powerset S:\)	\(\ds 0 \le \map \rho X \le \size X \)
\((\text R 5)\)	$:$	\(\ds \forall X, Y \in \powerset S:\)	\(\ds X \subseteq Y \implies \map \rho X \le \map \rho Y \)
\((\text R 6)\)	$:$	\(\ds \forall X, Y \in \powerset S:\)	\(\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y \)