Subband of Induced Operation is Set of Subbands

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 $\mathcal P \left({S}\right)$ of $S$

and
 * the operation $\circ_\mathcal P$ induced on $\mathcal P \left({S}\right)$ by $\circ$.

Let $T \subseteq \mathcal P \left({S}\right)$.

Let $\left({T, \circ_\mathcal P}\right)$ be a subband of $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$.

Then every element of $T$ is a subband of $\left({S, \circ}\right)$.

Case 1: $T$ is the Empty Set
By:
 * Empty Set is Submagma of Magma
 * Restriction of Associative Operation is Associative
 * Restriction of Idempotent Operation is Idempotent

it follows that $\left({\varnothing, \circ_\mathcal P}\right)$ is a subband of $\left({T, \circ_\mathcal P}\right)$.

Let $X \in \varnothing$.

Then by the definition of the empty set it follows that $\left({X, \circ}\right)$ is a subband of $\left({S, \circ}\right)$ vacuously.

Case 2: $T$ is Non-Empty
Let $X \in T$.

Then by definition of a subband $X$ is idempotent under $\circ_\mathcal P$.

That is:


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

By Subband iff Idempotent under Induced Operation we have that $\left({X, \circ}\right)$ is a subband of $\left({S, \circ}\right)$.

Hence the result.