Operation is Isomorphic to Power Structure Operation on Set of Singleton Subsets

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Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Let $S'$ denote the set of singleton elements of $\powerset S$.


Then $\struct {S, \circ}$ is isomorphic to $\struct {S', \circ_\PP}$.


Proof

Let $\phi: S \to S'$ be the mapping defined as:

$\forall x \in S: \map \phi x = \set x$

We have that:

\(\ds \forall a, b \in S: \, \) \(\ds \map \phi a\) \(=\) \(\ds \map \phi b\)
\(\ds \leadsto \ \ \) \(\ds \set a\) \(=\) \(\ds \set b\) Definition of $\phi$
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds b\) Singleton Equality

demonstrating that $\phi$ is an injection.

Then we have:

\(\ds \forall A \in S': \exists a \in S: \, \) \(\ds A\) \(=\) \(\ds \set a\) Definition of Singleton
\(\ds \leadsto \ \ \) \(\ds A\) \(=\) \(\ds \map \phi a\) Definition of $\phi$

demonstrating that $\phi$ is a surjection.

Hence by definition $\phi$ is a bijection.


By the definition of a magma, $S$ is closed under $\circ$.

That is:

$\forall a, b \in S, a \circ b \in S$

Hence:

$a \circ b \in \Dom \phi$

Also, by Power Structure of Magma is Magma, $S'$ is closed under $\circ_\PP$.

Hence:

$\set a \circ_\PP \set b \in S'$


Now:

\(\ds \forall a, b \in S: \, \) \(\ds \map \phi {a \circ b}\) \(=\) \(\ds \set {a \circ b}\) Definition of $\phi$
\(\ds \) \(=\) \(\ds \set a \circ_\PP \set b\) Definition of Subset Product
\(\ds \) \(=\) \(\ds \map \phi a \circ_\PP \map \phi b\) Definition of $\phi$

That is, $\phi$ is a homomorphism.


So $\phi$ is a bijective homomorphism.

Hence the result by definition of isomorphism.

$\blacksquare$


Sources

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