Condition for Mapping between Structures to be Homomorphism

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Theorem

Let $\struct {A, \odot}$ and $\struct {B, \circledast}$ be magmas.

Let $\struct {A \times B, \otimes}$ be the external direct product of $\struct {A, \odot}$ and $\struct {B, \circledast}$.


Let $\phi: A \to B$ be a mapping.

Let $\phi$ be considered as a subset of the Cartesian product $A \times B$.

Then:

$\phi$ is a homomorphism

if and only if:

the algebraic structure $\struct {\phi, \otimes_\phi}$ is a submagma of $\struct {A \times B, \otimes}$.


Proof

Let $\phi$ be a homomorphism

Let $\tuple {a, b}, \tuple {c, d} \in A \times B$ such that:

\(\ds \tuple {a, b}\) \(\in\) \(\ds \phi\)
\(\ds \tuple {c, d}\) \(\in\) \(\ds \phi\)

Then:

\(\ds \map \phi a\) \(=\) \(\ds b\) Definition of Mapping
\(\, \ds \land \, \) \(\ds \map \phi c\) \(=\) \(\ds d\)
\(\ds \leadstoandfrom \ \ \) \(\ds \map \phi {a \odot c}\) \(=\) \(\ds b \circledast d\) as $\phi$ is a homomorphism
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {a \odot c, b \circledast d}\) \(\in\) \(\ds \phi\) Definition of Mapping
\(\ds \leadstoandfrom \ \ \) \(\ds \tuple {a, b} \otimes_\phi \tuple {c, d}\) \(\in\) \(\ds \phi\) Definition of External Direct Product

That is:

$\struct {\phi, \otimes_\phi}$ is a closed subset of $\struct {A \times B, \otimes}$

which means the same thing as:

$\struct {\phi, \otimes_\phi}$ is a submagma of $\struct {A \times B, \otimes}$.

$\Box$


The argument reverses, so:

if $\struct {\phi, \otimes_\phi}$ is a submagma of $\struct {A \times B, \otimes}$

then:

$\phi$ is a homomorphism.

$\blacksquare$


Sources

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