Internal Direct Product/Examples/Non-Example 1

Example of External Direct Product which is not Internal Direct Product
Let $m$ and $n$ be integers such that $m, n > 1$.

Let $S$ be a set with $n$ elements.

Let $A$ and $B$ be subsets of $S$ which have $m$ and $n$ elements respectively.

Let $\struct {S, \gets}$ be the algebraic structure formed from $S$ with the left operation.

Then:
 * $\struct {S, \gets}$ is isomorphic with the external direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$

but:
 * $\struct {S, \gets}$ is not the internal direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$.

Proof
From Cardinality of Cartesian Product of Finite Sets, $S$ has the same number of elements as the Cartesian product of $A$ and $B$.

That is:
 * $\card {\struct {S, \gets} } = \card {\struct {A, \gets_A} \times \struct {B, \gets_B} }$

Hence by definition of cardinality, there exists a bijection between $S$ and $A \times B$.

Indeed, from Cardinality of Set of Bijections, there are $m n!$ such bijections.

First we demonstrate that $\struct {S, \gets}$ is isomorphic with the external direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$.

Let $\phi: A \times B \to S$ be an arbitrary bijection.

We have:

demonstrating isomorphism.

Let $\phi: A \times B \to S$ be the mapping defined as:
 * $\map \phi {a, b} = a \gets b$

Let $\tuple {a, b}$ and $\tuple {c, b}$ be arbitrary elements of $A \times B$ such that $a \ne c$.

As the cardinality of $A$ is greater than $1$, it is apparent that this is possible.

Thus:
 * $\tuple {a, b} \ne \tuple {c, b}$

But we have:

demonstrating that $\phi$ is not an injection.

Thus $\phi$ is not a bijection.

Hence by definition $\phi$ is not an isomorphism.

It follows that there can be no isomorphism from $\struct {A, \gets_A} \times \struct {B, \gets_B}$ to $\struct {S, \gets}$.

That is, $\struct {S, \gets}$ is not the internal direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$.