\begin{definition}[Definition:Closure Axioms (Matroid)]
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''' {{iff}}:
{{begin-axiom}}
{{axiom | n = \text S 1
        | q = \forall X  \in \powerset S
        | mr = X \subseteq \map \sigma X
}}
{{axiom | n = \text S 2
        | q = \forall X, Y \in \powerset S
        | mr = X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y
}}
{{axiom | n = \text S 3
        | q = \forall X \in \powerset S
        | mr = \map \sigma X = \map \sigma {\map \sigma X}
}}
{{axiom | n = \text S 4
        | q = \forall X \in \powerset S \land x, y \in S
        | mr = y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y}
}}
{{end-axiom}}
\end{definition}