Internal Direct Product/Examples

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Examples of Internal Direct Products

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 $\struct {B, \gets_B}$.