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:

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

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 $$\circ_2$$.

Some authors refer to this as the cartesian product of $$\left({S, \circ_1}\right)$$ and $$\left({T, \circ_2}\right)$$. Others (whose expositions are not concerned with the Internal Direct Product) call it just the direct product.