User:Lord Farin/Sandbox/Magma of Sets

= Definition:Magma of Sets =

Definition
Let $X$ be a set, and let $\mathcal S \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Let $I$ be an index set.

For every $i \in I$, let $\mathcal{D}_i \subseteq \mathcal P \left({X}\right)^{J_i}$, where $J_i$ is also an index set.

Let $\phi_i : \mathcal{D}_i \to \mathcal P \left({X}\right)$ be a mapping, for each $i \in I$.

Then $\mathcal S$ is a magma of sets for $\left\{{\phi_i : i \in I}\right\}$ iff:


 * $\forall i \in I: \left({S_{j_i}}\right)_{j_i \in J_i} \in \mathcal{S}^{J_i} \cap \mathcal{D}_i \implies \phi_i \left({\left({S_{j_i}}\right)_{j_i \in J_i}}\right) \in \mathcal S$

that is, iff $\mathcal S$ is closed under $\phi_i$ for all $i \in I$.

Examples
$\sigma$-algebra, ring of sets, Dynkin system, topology, monotone class, power set (nice how that one rolls out intrinsically)...

Also see

 * Magma of Sets Generated by Collection of Subsets
 * Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets