Subband of Induced Operation is Set of Subbands

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

Let $\struct {\powerset S, \circ_\PP}$ be the algebraic structure consisting of:
 * the power set $\powerset S$ of $S$

and
 * the operation $\circ_\PP$ induced on $\powerset S$ by $\circ$.

Let $T \subseteq \powerset S$.

Let $\struct {T, \circ_\PP}$ be a subband of $\struct {\powerset S, \circ_\PP}$.

Then every element of $T$ is a subband of $\struct {S, \circ}$.

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 $\struct {\O, \circ_\PP}$ is a subband of $\struct {T, \circ_\PP}$.

Let $X \in \O$.

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

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

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

That is:


 * $X \circ_\PP X = X$

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

Hence the result.