Definition:Closure Axioms (Matroid)

From ProofWiki
Jump to navigation Jump to search


Let $S$ be a finite set.

Let $\sigma : \powerset S \to \powerset S$ be a mapping where $\powerset S$ denotes the power set of $S$.

$\sigma$ is said to satisfy the closure axioms if and only if:

\((\text S 1)\)   $:$     \(\ds \forall X \in \powerset S:\) \(\ds X \subseteq \map \sigma X \)      
\((\text S 2)\)   $:$     \(\ds \forall X, Y \in \powerset S:\) \(\ds X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y \)      
\((\text S 3)\)   $:$     \(\ds \forall X \in \powerset S:\) \(\ds \map \sigma X = \map \sigma {\map \sigma X} \)      
\((\text S 4)\)   $:$     \(\ds \forall X \in \powerset S \land x, y \in S:\) \(\ds y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y} \)