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 $X$ if and only if:
- $\forall i \in I: \map {\phi_i} {\family {S_j}_{j \mathop \in J_i} } \in \SS$
for every indexed family $\family {S_j}_{j \mathop \in J_i} \in \SS^{J_i}$ in the domain of $\phi$.
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That is, if and only if $\SS$ is closed under $\phi_i$ for all $i \in I$.
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Examples
ring of sets, Dynkin system, monotone class, subgroup, normal subgroup (include the conjugation operations)
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Historical Note
The term magma of sets was specifically coined by the $\mathsf{Pr} \infty \mathsf{fWiki}$ user Lord_Farin to accommodate this concept.
No other references to structures this general have been located in the literature as of yet.