Axiom: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.
Formulation 1
$\rho$ is said to satisfy the rank axioms if and only if:
\((\text R 1)\) | $:$ | \(\ds \map \rho \O = 0 \) | |||||||
\((\text R 2)\) | $:$ | \(\ds \forall X \in \powerset S \land y \in S:\) | \(\ds \map \rho X \le \map \rho {X \cup \set y} \le \map \rho X + 1 \) | ||||||
\((\text R 3)\) | $:$ | \(\ds \forall X \in \powerset S \land y, z \in S:\) | \(\ds \map \rho {X \cup \set y} = \map \rho {X \cup \set z} = \map \rho X \implies \map \rho {X \cup \set y \cup \set z} = \map \rho X \) |
Formulation 2
$\rho$ is said to satisfy the rank axioms if and only if:
\((\text R 4)\) | $:$ | \(\ds \forall X \in \powerset S:\) | \(\ds 0 \le \map \rho X \le \size X \) | ||||||
\((\text R 5)\) | $:$ | \(\ds \forall X, Y \in \powerset S:\) | \(\ds X \subseteq Y \implies \map \rho X \le \map \rho Y \) | ||||||
\((\text R 6)\) | $:$ | \(\ds \forall X, Y \in \powerset S:\) | \(\ds \map \rho {X \cup Y} + \map \rho {X \cap Y} \le \map \rho X + \map \rho Y \) |