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 if and only if $\struct {X, \circ}$ is a subband of $\struct {S, \circ}$.

Proof

Subbandhood implies Idempotency

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:

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

$\Box$

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.

$\blacksquare$