Identity Element for Operation Induced on Power Set

Theorem
Let $\struct {S, \circ}$ be a magma whose underlying set $S$ is non-empty.

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

Then:
 * a subset $J$ of $S$ is an identity element of the algebraic structure $\struct {\powerset S, \circ_\PP}$


 * there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.
 * there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.

Sufficient Condition
Let $J \subseteq S$ such that $J$ is an identity element of $\struct {\powerset S, \circ_\PP}$.

We have:

That is, the elements of $J$ are all left identities for $\circ$.

Similarly:

That is, the elements of $J$ are all right identities for $\circ$.

So we have established that $J$ contains at least one element which is both a left identity and a right identity.

From Left and Right Identity are the Same, there is only one element of $J$, and it is the necessarily unique identity element for $\circ$.

That is:
 * there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.

Necessary Condition
Let $\struct {S, \circ}$ have an identity element $e$.

Let $J = \set e$.

Then we have:

and similarly:

So:
 * $\forall A \in \powerset S: A \circ_\PP J = A = J \circ_\PP A$

and it is seen that $J$ is an identity element for $\circ_\PP$.