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 $J_i$ be an index set, and let:


 * $\phi_i: \mathcal P \left({X}\right)^{J_i} \to \mathcal P \left({X}\right)$

be a partial mapping.

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


 * $\forall i \in I: \phi_i \left({\left({S_j}\right)_{j \in J_i}}\right) \in \mathcal S$

for every $\left({S_j}\right)_{j \in J_i} \in \mathcal S^{J_i}$ in the domain of $\phi$.

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

Examples

 * $\sigma$-Algebra as Magma of Sets
 * Topology as Magma of Sets

ring of sets, Dynkin system, monotone class, subgroup, normal subgroup (include the conjugation operations)

Historical Note
The name magma of sets was specifically invented by the ProofWiki user Lord_Farin to accommodate for this concept. No other references to structures this general have been located in the literature as of yet.

Also see

 * Power Set is Magma of Sets
 * Intersection of Magmas of Sets is Magma of Sets
 * Magma of Sets Generated by Collection of Subsets
 * Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets