Definition:Magma of Sets Generated by Collection of Subsets

Definition
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.

Also see

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