Subband iff Idempotent under Induced Operation

Theorem
Let $\left({S, \circ}\right)$ be a band.

Let $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\mathcal P \left({S}\right)$ by $\circ$.

Let $X \in \mathcal P \left({S}\right)$.

Then $X$ is idempotent iff $\left({X, \circ}\right)$ is a subband of $\left({S, \circ}\right)$.

Proving $\left({X \circ_{\mathcal P} X}\right) \subseteq X$
Let $c \in X \circ_{\mathcal P} X$.

By the definition of subset product for some $a, b \in X$ we have:


 * $a \circ b = c$

Suppose $c \notin X$.

Then:


 * $a \circ b \notin X$

Which contradicts that $\left({X, \circ}\right)$ is a subband.

Proving $X \subseteq \left({X \circ_{\mathcal P} X}\right)$
Let $a \in X$.

By the definition of subset product:


 * $X \circ_{\mathcal P} X = \{ a \circ b: a, b \in X \}$

As $\circ$ is idempotent:


 * $a \circ a = a$.

Thus:


 * $a \in \left({X \circ_{\mathcal P} X}\right)$

Hence by the definition of subset:


 * $X \subseteq \left({X \circ_{\mathcal P} X}\right)$

Idempotency implies Subbandhood
Let $X \in \mathcal P \left({S}\right)$.

Suppose $X$ is idempotent:

That is suppose:


 * $X \circ_{\mathcal P} X = X$

Let $a, b \in X$.

By the definition of subset product:


 * $X \circ_{\mathcal P} X = \{ a \circ b: a, b \in X \}$

Then $a \circ b \in X$.

Hence $\left({X, \circ}\right)$ is a magma.

By Restriction of Operation Associativity it is a semigroup.

Finally by Restriction of Operation Idempotency it is a band.