Internal Direct Product/Examples/Non-Example 1

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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:

\(\ds \forall \tuple {a, b}, \tuple {c, d} \in A \times B: \, \) \(\ds \) \(\) \(\ds \map \phi {\tuple {a, b} \gets \tuple {c, d} }\)
\(\ds \) \(=\) \(\ds \map \phi {a \gets c, b \gets d}\) Definition of External Direct Product
\(\ds \) \(=\) \(\ds \map \phi {c, d}\) Definition of Left Operation
\(\ds \) \(=\) \(\ds \map \phi {a, b} \gets \map \phi {c, d}\) Definition of Left Operation

demonstrating isomorphism.

$\Box$


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:

\(\ds \map \phi {a, b}\) \(=\) \(\ds a \gets b\) Definition of $\phi$
\(\ds \) \(=\) \(\ds b\) Definition of $\gets$
\(\ds \) \(=\) \(\ds c \gets b\) Definition of $\gets$
\(\ds \) \(=\) \(\ds \map \phi {c, b}\) Definition of $\phi$
\(\ds \leadsto \ \ \) \(\ds \map \phi {a, b}\) \(=\) \(\ds \map \phi {c, b}\)

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 $\struct {B, \gets_B}$.

$\blacksquare$


Sources

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