# 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 $\phi$.

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

Although this article appears correct, it's inelegant. There has to be a better way of doing it.Dispose of the index set to prevent problems on Topology as Magma of Sets, where $I$ would be a proper classYou can help Proof Wiki by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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.