# 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
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.

$\blacksquare$