Definition:Magma of Sets

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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 $X$ if and only if:

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

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


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

Examples

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

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.


Also see