Definition:Magma of Sets Generated by Collection of Subsets
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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.