Equivalent Conditions for Entropic Structure/Mapping from External Direct Product is Homomorphism
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Theorem
Let $\struct {S, \odot}$ be an algebraic structure.
Let $\struct {S \times S, \otimes}$ denote the external direct product of $\struct {S, \odot}$ with itself:
- $\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \tuple {x_1, y_1} \otimes \tuple {x_2, y_2} = \tuple {x_1 \odot x_2, y_1 \odot y_2}$
Consider the operation $\odot$ as a mapping from $S \times S$ to $S$.
That is:
- $\forall a, b \in S: \map \odot {a, b} = a \odot b$
Then:
- $\odot: S \times S \to S$ is a homomorphism from $\struct {S \times S, \otimes}$ to $\struct {S, \odot}$
- $\struct {S, \odot}$ is an entropic structure.
Proof
Sufficient Condition
Let $\struct {S, \odot}$ be such that $\odot: S \times S \to S$ is a homomorphism.
Let $\tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S$ be arbitrary.
We have:
\(\ds \) | \(\) | \(\ds \paren {x_1 \odot x_2} \odot \paren {y_1 \odot y_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map \odot {x_1, x_2} } \odot \paren {\map \odot {y_1, y_2} }\) | Definition of $\odot$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \odot {\tuple {x_1, x_2} \otimes \tuple {y_1, y_2} }\) | by hypothesis $\odot$ is a homomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \odot {\tuple {x_1 \odot y_1, x_2 \odot y_2} }\) | Definition of External Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x_1 \odot y_1} \odot \paren {x_2 \odot y_2}\) | Definition of $\odot$ |
and it is seen $\struct {S, \odot}$ is an entropic structure by definition.
$\Box$
Necessary Condition
Let $\struct {S, \odot}$ be an entropic structure.
Then:
\(\ds \forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \, \) | \(\ds \) | \(\) | \(\ds \map \odot {\tuple {x_1, x_2} \otimes \tuple {y_1, y_2} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \odot {\tuple {x_1 \odot y_1, x_2 \odot y_2} }\) | Definition of External Direct Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x_1 \odot y_1} \odot \paren {x_2 \odot y_2}\) | Definition of $\odot$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x_1 \odot x_2} \odot \paren {y_1 \odot y_2}\) | Definition of Entropic Structure | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map \odot {x_1, x_2} } \odot \paren {\map \odot {y_1, y_2} }\) | Definition of $\odot$ |
Hence by definition $\odot: S \times S \to S$ is a homomorphism.
$\blacksquare$
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: Exercise $13.13$