Equivalence of Definitions of Matroid Rank Axioms

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Theorem

Let $S$ be a finite set.

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


The following definitions for the Rank Axioms are equivalent:

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

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


Proof

Formulation 1 implies Formulation 2

Follows immediately from:

$\Box$


Formulation 2 implies Formulation 1

Let $\rho$ satisfy formulation 2 of the rank axioms:

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



$\rho$ satisfies $(\text R 1)$

We have:

\(\ds 0\) \(\le\) \(\ds \map \rho \O\) Rank axiom $(\text R 4)$
\(\ds \) \(\le\) \(\ds \card \O\) Rank axiom $(\text R 4)$
\(\ds \) \(=\) \(\ds 0\) Cardinality of Empty Set

Hence:

$\map \rho \O = 0$

$\Box$


$\rho$ satisfies $(\text R 2)$

Let $X \subseteq S$.

Let $y \in S$.


We have:

\(\ds \map \rho X\) \(\le\) \(\ds \map \rho {X \cup y}\) Rank axiom $(\text R 5)$
\(\ds \) \(\le\) \(\ds \map \rho X + \map \rho {\set y} - \map \rho {X \cup \set y}\) Rank axiom $(\text R 6)$
\(\ds \) \(\le\) \(\ds \map \rho X + \map \rho {\set y}\)
\(\ds \) \(\le\) \(\ds \map \rho X + \card {\set y}\) Rank axiom $(\text R 4)$
\(\ds \) \(\le\) \(\ds \map \rho X + 1\) Cardinality of Singleton

This proves rank axiom $(\text R 2)$

$\Box$


$\rho$ satisfies $(\text R 3)$

Let $X \subseteq S$.

Let $x, y \in S$.

Let $\map \rho {X \cup \set x} = \map \rho {X \cup \set y} = \map \rho X$.


We have:

\(\ds \map \rho X\) \(\le\) \(\ds \map \rho {X \cup \set x \cup \set y }\) Rank axiom $(\text R 5)$
\(\ds \) \(=\) \(\ds \map \rho {\paren{X \cup \set x} \cup \paren{X \cup \set y} }\)
\(\ds \) \(\le\) \(\ds \map \rho {X \cup \set x} + \map \rho {X \cup \set x} - \map \rho {\paren{X \cup \set x} \cap \paren{X \cup \set y} }\) Rank axiom $(\text R 6)$
\(\ds \) \(\le\) \(\ds \map \rho {X \cup \set x} + \map \rho {X \cup \set x} - \map \rho X\)
\(\ds \) \(=\) \(\ds \map \rho X + \map \rho X - \map \rho X\)
\(\ds \) \(=\) \(\ds \map \rho X\)

Hence:

$\map \rho {X \cup \set x \cup \set y} = \map \rho X$

This proves rank axiom $(\text R 3)$

$\blacksquare$


Sources

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