Subband iff Idempotent under Induced Operation

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

Let $\struct {\powerset S, \circ_\PP}$ be the algebraic structure consisting of the power set of $S$ and the operation induced on $\powerset S$ by $\circ$.

Let $X \in \powerset S$.

Then $X$ is idempotent $\struct {X, \circ}$ is a subband of $\struct {S, \circ}$.

Proving $\paren {X \circ_PP X} \subseteq X$
Let $c \in X \circ_\PP X$.

By 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 $\struct {X, \circ}$ is a subband.

Proving $X \subseteq \paren {X \circ_\PP X}$
Let $a \in X$.

By definition of subset product:


 * $X \circ_\PP X = \set {a \circ b: a, b \in X}$

As $\circ$ is idempotent:


 * $a \circ a = a$.

Thus:


 * $a \in \set {X \circ_\PP X}$

Hence by definition of subset:


 * $X \subseteq \paren {X \circ_\PP X}$

Idempotency implies Subbandhood
Let $X \in \powerset S$.

Suppose $X$ is idempotent:

That is suppose:


 * $X \circ_\PP X = X$

Let $a, b \in X$.

By the definition of subset product:


 * $X \circ_\PP X = \set {a \circ b: a, b \in X}$

Then $a \circ b \in X$.

Hence $\struct {X, \circ}$ is a magma.

By Restriction of Associative Operation is Associative it is a semigroup.

Finally by Restriction of Idempotent Operation is Idempotent it is a band.