Definition:Magma of Sets

Definition
Let $X$ be a set.

Let $\SS \subseteq \powerset X$ be a set of subsets of $X$.

Let $I$ be an index set.

For every $i \in I$, let $J_i$ be an index set, and let:


 * $\phi_i: \powerset X^{J_i} \to \powerset X$

be a partial mapping.

Then $\SS$ is a magma of sets for $\set {\phi_i: i \in I}$ on $X$ :


 * $\forall i \in I: \map {\phi_i} {\family {S_j}_{j \mathop \in J_i} } \in \SS$

for every indexed family $\family {S_j}_{j \mathop \in J_i} \in \SS^{J_i}$ in the domain of $\phi$.

That is, $\SS$ 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)

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