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 $$\left({S, \circ}\right)$$.

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

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

The structure $$\left({S, \circ}\right)$$ is the internal direct product of $$\left \langle {S_n} \right \rangle$$ if the mapping:

$$C: \prod_{k=1}^n S_k \to S: C \left({s_1, \ldots, s_n}\right) = \prod_{k=1}^n s_k$$

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