Subband of Induced Operation is Set of Subbands

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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 $\struct {S, \circ}$.


Proof

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.

$\blacksquare$