Definition:Magma of Sets Generated by Collection of Subsets

Definition
Let $X$ be a set, and let $\Phi := \left\{{\phi_i: i \in I}\right\}$ be a collection of partial mappings with codomain $\mathcal P \left({X}\right)$, the power set of $X$.

Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$.

Then the magma of sets for $\Phi$ generated by $\mathcal G$ is the unique magma of sets $\mathcal S \subseteq \mathcal P \left({X}\right)$ satisfying:


 * $(1): \quad \mathcal G \subseteq \mathcal S$
 * $(2): \quad \mathcal G \subseteq \mathcal T$ implies that $\mathcal S \subseteq \mathcal T$ for every magma of sets $\mathcal T$

To speak of the unique magma of sets generated by $\mathcal G$ is justified by Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets.

Also see

 * Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets