# 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.In particular: Dispose of the index set to prevent problems on Topology as Magma of Sets, where $I$ would be a proper classYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ 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)

Work In ProgressIn particular: Make rigid by providing pagesYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing 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 `{{WIP}}` from the code. |

## 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.