Category:Formulation 1 Rank Axioms Implies Rank Function of Matroid

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

Let $\rho$ satisfy formulation 1 of the rank axioms:

$(\text R 1)$	$:$		$\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 $

Then $\rho$ is the rank function of a matroid on $S$.

Pages in category "Formulation 1 Rank Axioms Implies Rank Function of Matroid"

The following 13 pages are in this category, out of 13 total.

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