Subband of Induced Operation is Set of Subbands

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Let $\left({S, \circ}\right)$ be a band.

Let $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$ be the algebraic structure consisting of:

the power set $\mathcal P \left({S}\right)$ of $S$


the operation $\circ_\mathcal P$ induced on $\mathcal P \left({S}\right)$ by $\circ$.

Let $T \subseteq \mathcal P \left({S}\right)$.

Let $\left({T, \circ_\mathcal P}\right)$ be a subband of $\left({\mathcal P \left({S}\right), \circ_\mathcal P}\right)$.

Then every element of $T$ is a subband of $\left({S, \circ}\right)$.


Case 1: $T$ is the Empty Set


Empty Set is Submagma of Magma
Restriction of Associative Operation is Associative
Restriction of Idempotent Operation is Idempotent

it follows that $\left({\varnothing, \circ_\mathcal P}\right)$ is a subband of $\left({T, \circ_\mathcal P}\right)$.

Let $X \in \varnothing$.

Then by the definition of the empty set it follows that $\left({X, \circ}\right)$ is a subband of $\left({S, \circ}\right)$ vacuously.


Case 2: $T$ is Non-Empty

Let $X \in T$.

Then by definition of a subband $X$ is idempotent under $\circ_\mathcal P$.

That is:

$X \circ_\mathcal P X = X$

By Subband iff Idempotent under Induced Operation we have that $\left({X, \circ}\right)$ is a subband of $\left({S, \circ}\right)$.

Hence the result.