# Category:Axioms/Matroid Rank Axioms

This category contains axioms related to Matroid Rank Axioms.

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:

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

## Pages in category "Axioms/Matroid Rank Axioms"

The following 3 pages are in this category, out of 3 total.