Category:Definitions/Direct Products

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This category contains definitions related to Direct Products.
Related results can be found in Category:Direct Products.

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:

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