Definition:Operation Induced by Direct Product

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Definition

Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures.

Let $S = S_1 \times S_2$ be the cartesian product of $S_1$ and $S_2$.


Then the operation induced on $S$ by $\circ_1$ and $\circ_2$ is the operation $\circ$ defined as:

$\tuple {s_1, s_2} \circ \tuple {t_1, t_2} := \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2}$

for all ordered pairs in $S$.


General Definition

Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ be algebraic structures.

Let $\ds \SS_n = \prod_{k \mathop = 1}^n S_k$ be the cartesian product of $S_1, S_2, \ldots, S_n$.


Then the operation induced on $\SS_n$ by $\circ_1, \ldots, \circ_n$ is the operation $\circledcirc_n$ defined as:

$\tuple {s_1, s_2, \ldots, s_n} \circledcirc_n \tuple {t_1, t_2, \ldots, t_n} := \begin{cases} s_1 \circ_1 t_1 & : n = 1 \\ \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2} & : n = 2 \\ \tuple {\tuple {s_1, s_2, \ldots, s_{n - 1} } \circledcirc_{n - 1} \tuple {t_1, t_2, \ldots, t_{n - 1} }, s_n \circ_n t_n} & : n > 2 \end{cases}$

for all ordered $n$-tuples in $\SS_n$.


That is:

$\tuple {s_1, s_2, \ldots, s_n} \circledcirc_n \tuple {t_1, t_2, \ldots, t_n} := \tuple {s_1 \circ_1 t_1, s_2 \circ_2 t_2, \ldots, s_n \circ_n t_n}$


Also see


Sources