Definition:Internal Direct Product
Definition
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {A, \circ {\restriction_A} }$ and $\struct {B, \circ {\restriction_B} }$ be closed algebraic substructures of $\struct {S, \circ}$, where $\circ {\restriction_A}$ and $\circ {\restriction_B}$ are the operations induced by the restrictions of $\circ$ to $A$ and $B$ respectively.
$\struct {S, \circ}$ is the internal direct product of $A$ and $B$ if and only if:
- the mapping $\phi: A \times B \to S$ defined as:
- $\forall a \in A, b \in B: \map \phi {a, b} = a \circ b$
- is an isomorphism from the (external) direct product $\struct {A, \circ {\restriction_A} } \times \struct {B, \circ {\restriction_B} }$ onto $\struct {S, \circ}$.
General Definition
Let $\struct {S, \circ}$ be an algebraic structure with $1$ operation.
Let $\struct {S_1, \circ {\restriction_{S_1} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ be closed algebraic substructures of $\struct {S, \circ}$, 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.
$\struct {S, \circ}$ is the internal direct product of $\sequence {S_n}$ if and only if:
- the mapping $\ds \phi: \prod_{k \mathop = 1}^n S_k \to S$ defined as:
- $\ds \forall i \in \set {1, 2, \ldots, n}: \forall s_i \in S_i: \map \phi {s_1, \ldots, s_n} = \prod_{k \mathop = 1}^n s_k$
- is an isomorphism from the (external) direct product $\struct {S_1, \circ {\restriction_{S_1} } } \times \cdots \times \struct {S_n, \circ {\restriction_{S_n} } }$ onto $\struct {S, \circ}$.
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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
The set of algebraic substructures $\struct {S_1, \circ {\restriction_{S_1} } }, \struct {S_2, \circ {\restriction_{S_2} } }, \ldots, \struct {S_n, \circ {\restriction_{S_n} } }$ whose (external) direct product is isomorphic with $\struct {S, \circ}$ is called a decomposition of $S$.
Also known as
Some sources refer to the internal direct product simply as the direct product.
Others call it the direct composite.
It is also commonplace for sources which evolve their expositions on abstract algebra purely from an arithmetical perspective to refer to the direct sum when the operation is, or derives from, addition.
Examples
External Direct Product which is not Internal Direct Product
Let $m$ and $n$ be integers such that $m, n > 1$.
Let $S$ be a set with $n$ elements.
Let $A$ and $B$ be subsets of $S$ which have $m$ and $n$ elements respectively.
Let $\struct {S, \gets}$ be the algebraic structure formed from $S$ with the left operation.
Then:
- $\struct {S, \gets}$ is isomorphic with the external direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$
but:
- $\struct {S, \gets}$ is not the internal direct product of $\struct {A, \gets_A}$ and $\struct {B, \gets_B}$.
Also see
- Definition:External Direct Product
- Definition:Internal Group Direct Product
- Definition:Ring Direct Sum
- Results about internal direct products can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces