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:


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

be a partial mapping.

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


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

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

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

Examples
ring of sets, Dynkin system, topology, monotone class, subgroup
 * $\sigma$-Algebra as Magma of Sets

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