# Definition:Closure Axioms (Matroid)

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## Definition

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:

\((\text S 1)\) | $:$ | \(\ds \forall X \in \powerset S:\) | \(\ds X \subseteq \map \sigma X \) | |||||

\((\text S 2)\) | $:$ | \(\ds \forall X, Y \in \powerset S:\) | \(\ds X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y \) | |||||

\((\text S 3)\) | $:$ | \(\ds \forall X \in \powerset S:\) | \(\ds \map \sigma X = \map \sigma {\map \sigma X} \) | |||||

\((\text S 4)\) | $:$ | \(\ds \forall X \in \powerset S \land x, y \in S:\) | \(\ds y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y} \) |

## Sources

- 1976: Dominic Welsh:
*Matroid Theory*... (previous) ... (next) Chapter $1.$ $\S 2.$ Axiom Systems for a Matroid: Theorem $4$