Definition:Internal Direct Product
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Definition
Let $\left({S_1, \circ \restriction_{S_1}}\right), \left({S_2, \circ \restriction_{S_2}}\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$
where $\circ \restriction_{S_1}, \circ \restriction_{S_2}$ are the operations induced by 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 $\left({S_1, \circ \restriction_{S_1}}\right) \times \left({S_2, \circ \restriction_{S_2}}\right)$ onto $\left({S, \circ}\right)$.
The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ \restriction_{S_1}$ and $\circ \restriction_{S_2}$.
It can be seen that the mapping $C$ is the restriction of the mapping $\circ$ of $S \times S$ to the subset $S_1 \times S_2$.
General Definition
Let $\left({S_1, \circ \restriction_{S_1}}\right), \ldots, \left({S_n, \circ \restriction_{S_n}}\right)$ be closed algebraic substructures of an algebraic structure $\left({S, \circ}\right)$
where $\circ \restriction_{S_1}, \ldots, \circ \restriction_{S_n}$ are the operations induced by 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:
- $\displaystyle C: \prod_{k=1}^n S_k \to S: C \left({s_1, \ldots, s_n}\right) = \prod_{k=1}^n s_k$
is an isomorphism from the cartesian product $\left({S_1, \circ \restriction_{S_1}}\right) \times \cdots \times \left({S_n, \circ \restriction_{S_n}}\right)$ onto $\left({S, \circ}\right)$.
The operation $\circ$ on $S$ is the operation induced on $S$ by $\circ \restriction_{S_1}, \circ \restriction_{S_2}, \ldots, \circ \restriction_{S_n}$.
Decomposition
Such a set of algebraic substructures $\left({S_1, \circ \restriction_{S_1}}\right), \left({S_2, \circ \restriction_{S_1}}\right)$ whose direct product forms $\left({S, \circ}\right)$ is called a decomposition of $S$.
Also known as
Some authors call this just the direct product. Some authors call it the direct composite.
Also see
- Results about internal direct products can be found here.
Sources
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 13$
