Definition:External Direct Product

Let $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$ be algebraic structures.

The external direct product $$\left({S \times T, \circ}\right)$$ of two algebraic structures $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$ is the set of ordered pairs:

$$\left({S \times T, \circ}\right) = \left\{{\left({s, t}\right): s \in S, t \in T}\right\}$$

where the operation $$\circ$$ is defined as: $$ \left({s_1, t_1}\right) \circ \left({s_2, t_2}\right) = \left({s_1 \circ_1 s_2, t_1 \circ_2 t_2}\right)$$

$$\circ$$ is the operation induced on $$S \times T$$ by $$\circ_1$$ and $$\circ_2$$.

Another notation sometimes seen for $$\left({S \times T, \circ}\right)$$ is $$\left({S \oplus T, \circ}\right)$$.

Generalized Definition
Let $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$$ be algebraic structures.

Let $$S = \prod_{k=1}^n S_k$$ be as defined in generalized cartesian product.

The operation induced on $$S$$ by $$\circ_1, \ldots, \circ_n$$ is the operation $$\circ$$ defined by:

$$\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$$

for all ordered $n$-tuples in $$S$$.

The algebraic structure $$\left({S, \circ}\right)$$ is called the (external) direct product of $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$$.

Structures with Two Operations
Let $$\left({S_1, +_1 ,\circ_1}\right), \left({S_2, +_2 ,\circ_2}\right), \ldots, \left({S_n, +_n ,\circ_n}\right)$$ be algebraic structures with two operations.

Let $$S = \prod_{k=1}^n S_k$$ be as defined in generalized cartesian product.


 * The operation induced on $$S$$ by $$+_1, \ldots, +_n$$ is the operation $$+$$ defined by:

$$\left({s_1, s_2, \ldots, s_n}\right) + \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 +_1 t_1, s_2 +_2 t_2, \ldots, s_n +_n t_n}\right)$$


 * The operation induced on $$S$$ by $$\circ_1, \ldots, \circ_n$$ is the operation $$\circ$$ defined by:

$$\left({s_1, s_2, \ldots, s_n}\right) \circ \left({t_1, t_2, \ldots, t_n}\right) = \left({s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}\right)$$

for all ordered $n$-tuples in S.