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 $\struct {S, \odot}$

if and only if:

$\struct {S, \odot}$ 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

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