Definition:Matroid Axioms

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Definition

Let $S$ be a finite set.

Let $\mathscr I$ be a set of subsets of $S$.


The matroid axioms are the conditions on $S$ and $\mathscr I$ in order for the ordered pair $\struct {S, \mathscr I}$ to be a matroid:

Axioms 1

\((\text I 1)\)   $:$   \(\displaystyle \O \in \mathscr I \)             
\((\text I 2)\)   $:$     \(\displaystyle \forall X \in \mathscr I: \forall Y \subseteq S:\) \(\displaystyle Y \subseteq X \implies Y \in \mathscr I \)             
\((\text I 3)\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size V < \size U \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I \)             


Axioms 2

\((\text I 1)\)   $:$   \(\displaystyle \O \in \mathscr I \)             
\((\text I 2)\)   $:$     \(\displaystyle \forall X \in \mathscr I: \forall Y \subseteq S:\) \(\displaystyle Y \subseteq X \implies Y \in \mathscr I \)             
\((\text I 3')\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size U = \size V + 1 \implies \exists x \in U \setminus V : V \cup \set x \in \mathscr I \)             


Axioms 3

\((\text I 1)\)   $:$   \(\displaystyle \O \in \mathscr I \)             
\((\text I 2)\)   $:$     \(\displaystyle \forall X \in \mathscr I: \forall Y \subseteq S:\) \(\displaystyle Y \subseteq X \implies Y \in \mathscr I \)             
\((\text I 3'')\)   $:$     \(\displaystyle \forall U, V \in \mathscr I:\) \(\displaystyle \size V < \size U \implies \exists Z \subseteq U \setminus V : \paren{V \cup Z \in \mathscr I} \land \paren{\size{V \cup Z} = \size U} \)             


Axioms 4

\((\text I 1)\)   $:$   \(\displaystyle \O \in \mathscr I \)             
\((\text I 2)\)   $:$     \(\displaystyle \forall X \in \mathscr I: \forall Y \subseteq S:\) \(\displaystyle Y \subseteq X \implies Y \in \mathscr I \)             
\((\text I 3''')\)   $:$     \(\displaystyle \forall A \subseteq S:\) \(\displaystyle \text{ all maximal subsets } Y \subseteq A \text{ with } Y \in \mathscr I \text{ have the same cardinality} \)             


Also see