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
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 $\struct {S, \circ}$.
Proof
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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.
$\Box$
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.
$\blacksquare$