Existence and Uniqueness of Magma of Sets Generated by Collection of Subsets
Theorem
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 generated by $\GG$ exists and is unique.
Proof
Uniqueness
Suppose that both $\SS$ and $\TT$ are magmas of sets for $\Phi$ generated by $\GG$.
Applying condition $(2)$ for these twice, we obtain:
- $\SS \subseteq \TT$
- $\TT \subseteq \SS$
By definition of set equality:
- $\SS = \TT$
$\Box$
Existence
Define $\SS$ by:
- $\SS := \bigcap \set {\TT: \GG \subseteq \TT}$
where $\TT$ ranges over the magmas of sets for $\Phi$ on $X$.
From Power Set is Magma of Sets, there intersection is at least one such $\TT$.
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From Intersection of Magmas of Sets is Magma of Sets, $\SS$ is a magma of sets for $\Phi$.
By Intersection Preserves Subsets: General Result: Corollary, $\GG \subseteq \SS$.
By Intersection is Subset: General Result, $\SS \subseteq \TT$ for every other magma of sets $\TT$ containing $\GG$.
Thus $\SS$ is a magma of sets for $\Phi$ generated by $\GG$.
$\blacksquare$