Definition:Internal Direct Product

Let $$\left({S_1, \circ|_{S_1}}\right), \left({S_2, \circ|_{S_2}}\right)$$ be closed algebraic substructures of an algebraic structure $$\left({S, \circ}\right)$$, where $$\circ|_{S_1}, \circ|_{S_2}$$ are the restrictions of $$\circ$$ to $$S_1, S_2$$ respectively.

The structure $$\left({S, \circ}\right)$$ is the internal direct product of $$S_1$$ and $$S_2$$ if the mapping:

$$C: S_1 \times S_2 \to S: C \left({\left({s_1, s_2}\right)}\right) = s_1 \circ s_2$$

is an isomorphism from the cartesian product $$\left({S_1, \circ|_{S_1}}\right) \times \left({S_2, \circ|_{S_2}}\right)$$ onto $$\left({S, \circ}\right)$$.

Some authors call this just the direct product.

It can be seen that the function $$C$$ is the restriction of the mapping $$\circ$$ of $$S \times S$$ to the subset $$S_1 \times S_2$$.