Definition:Rank Axioms (Matroid)

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Definition

Let $S$ be a finite set.

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


Definition 1

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

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


Definition 2

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

\((\text R 1')\)   $:$     \(\ds \forall X \in \powerset S:\) \(\ds 0 \le \map \rho X \le \size X \)             
\((\text R 2')\)   $:$     \(\ds \forall X, Y \in \powerset S:\) \(\ds X \subseteq Y \implies \map \rho X \le \map \rho Y \)             
\((\text R 3')\)   $:$     \(\ds \forall X, Y \in \powerset S:\) \(\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y \)