Power Structure of Subset is Closed iff Subset is Closed

Theorem
Let $\struct {S, \circ}$ be a magma.

Let $\circ_\PP$ be the operation induced on $\powerset S$, the power set of $S$.

Let $T \subseteq S$.

Then:
 * the algebraic structure $\struct {\powerset T, \circ_\PP}$ is closed


 * the algebraic structure $\struct {T, \circ}$ is closed.
 * the algebraic structure $\struct {T, \circ}$ is closed.

Sufficient Condition
Let $\struct {\powerset T, \circ_\PP}$ be closed.

Then:

That is, $\struct {T, \circ}$ is closed.

Necessary Condition
Let $\struct {T, \circ}$ be closed.

Then by definition $\struct {T, \circ}$ is a magma.

From Power Set of Magma under Induced Operation is Magma it follows that $\struct {\powerset T, \circ_\PP}$ is likewise a magma.

That is, the algebraic structure $\struct {\powerset T, \circ_\PP}$ is closed.