\begin{definition}[Definition:Magma of Sets Generated by Collection of Subsets]
Let $X$ be a set, and let $\Phi := \set {\phi_i: i \in I}$ be a collection of partial mappings with codomain $\powerset X$, the power set of $X$.
Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$.
Then the '''magma of sets for $\Phi$ generated by $\GG$''' is the unique magma of sets $\SS \subseteq \powerset X$ satisfying:
:$(1): \quad \GG \subseteq \SS$
:$(2): \quad \GG \subseteq \TT$ implies that $\SS \subseteq \TT$ for every magma of sets $\TT$
To speak of ''the unique'' '''magma of sets generated by $\GG$''' is justified by Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets.
\end{definition}