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.
$\blacksquare$