Equivalent Conditions for Entropic Structure/Pointwise Operation of Homomorphisms 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:

$\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \tuple {x_1, y_1} \otimes \tuple {x_2, y_2} = \tuple {x_1 \odot x_2, y_1 \odot y_2}$


Let $f$ and $g$ be mappings from $\struct {S \times S, \otimes}$ to $\struct {S, \odot}$.

Let $f \odot g$ denote the pointwise operation on $S^{S \times S}$ induced by $\odot$.


Then:

If $f$ and $g$ are homomorphisms, then $f \odot g$ is also a homomorphism

if and only if:

$\struct {S, \odot}$ is an entropic structure.


Proof

Sufficient Condition

Let $\struct {S, \odot}$ be such that if $f$ and $g$ are homomorphisms, then $f \odot g$ is also a homomorphism.

So, let $f: S \times S \to S$ and $g: S \times S \to S$ be arbitrary homomorphisms.

Let $w$, $x$, $y$ and $d$ in $S$ be arbitrary.

Then as $f$ and $g$ are arbitrary:

\(\ds \exists \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \, \) \(\ds \map f {x_1, y_1}\) \(=\) \(\ds w\)
\(\, \ds \land \, \) \(\ds \map f {x_2, y_2}\) \(=\) \(\ds x\)
\(\ds \map g {x_1, y_1}\) \(=\) \(\ds y\)
\(\, \ds \land \, \) \(\ds \map g {x_2, y_2}\) \(=\) \(\ds z\)


Then we have:

\(\ds \) \(\) \(\ds \paren {w \odot x} \odot \paren {y \odot z}\)
\(\ds \) \(=\) \(\ds \paren {\map f {x_1, y_1} \odot \map f {x_2, y_2} } \odot \paren {\map g {x_1, y_1} \odot \map g {x_2, y_2} }\) a priori
\(\ds \) \(=\) \(\ds \paren {\map f {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} } } \odot \paren {\map g {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} } }\) as $f$ and $g$ are both homomorphisms
\(\ds \) \(=\) \(\ds \map {\paren {f \odot g} } {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} }\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \map {\paren {f \odot g} } {x_1, y_1} \odot \map {\paren {f \odot g} } {x_2, y_2}\) as $f \odot g$ is a homomorphism
\(\ds \) \(=\) \(\ds \paren {w \odot y} \odot \paren {x \odot z}\) Definition of Pointwise Operation

As $w$, $x$, $y$ and $z$ are arbitrary, $\struct {S, \odot}$ is an entropic structure.

$\Box$


Necessary Condition

Let $\struct {S, \odot}$ be an entropic structure.

Let $f: S \times S \to S$ and $g: S \times S \to S$ be arbitrary homomorphisms.

Then:

\(\ds \forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \, \) \(\ds \) \(\) \(\ds \map {\paren {f \odot g} } {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} }\)
\(\ds \) \(=\) \(\ds \map f {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} } \odot \map g {\tuple {x_1, y_1} \otimes \tuple {x_2, y_2} }\) Definition of Pointwise Operation
\(\ds \) \(=\) \(\ds \paren {\map f {x_1, y_1} \odot \map f {x_2, y_2} } \odot \paren {\map g {x_1, y_1} \odot \map g {x_2, y_2} }\) as $f$ and $g$ are homomorphisms
\(\ds \) \(=\) \(\ds \paren {\map f {x_1, y_1} \odot \map g {x_1, y_1} } \odot \paren {\map f {x_2, y_2} \odot \map g {x_2, y_2} }\) Definition of Entropic Structure
\(\ds \) \(=\) \(\ds \paren {\map {\paren {f \odot g} } {x_1, y_1} } \odot \paren {\map {\paren {f \odot g} } {x_2, y_2} }\) Definition of Pointwise Operation

As $w$, $x$, $y$ and $z$ are arbitrary, $\struct {S, \odot}$ is an entropic structure.

$\blacksquare$


Sources

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