Equivalent Conditions for Entropic Structure/Pointwise Operation is Homomorphism
Theorem
Let $\struct {S, \odot}$ be an algebraic structure.
Let $\struct {T, \circledast}$ be an arbitrary algebraic structure.
Let $f$ and $g$ be mappings from $\struct {T, \circledast}$ to $\struct {S, \odot}$.
Let $f \odot g$ denote the pointwise operation on $S^T$ induced by $\odot$.
Then:
- If $f$ and $g$ are homomorphisms, then $f \odot g$ is also a homomorphism
- $\struct {S, \odot}$ 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 $f: T \to S$ and $g: T \to S$ be arbitrary homomorphisms.
Let $a, b \in T$ be arbitrary.
Because $T$ is arbitrary, and $f$ and $g$ are arbitrary, it follows that:
\(\ds \forall w, x, y, z \in S: \exists a, b \in T: \, \) | \(\ds \map f a\) | \(=\) | \(\ds w\) | |||||||||||
\(\, \ds \land \, \) | \(\ds \map f b\) | \(=\) | \(\ds x\) | |||||||||||
\(\ds \map g a\) | \(=\) | \(\ds y\) | ||||||||||||
\(\, \ds \land \, \) | \(\ds \map g b\) | \(=\) | \(\ds z\) |
Thus we have:
\(\ds \paren {w \odot x} \odot \paren {y \odot z}\) | \(=\) | \(\ds \paren {\map f a \odot \map f b} \odot \paren {\map g a \odot \map g b}\) | a priori | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f {a \circledast b} }\odot \paren {\map g {a \circledast b} }\) | as $f$ and $g$ are both homomorphisms | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f \odot g} } {a \circledast b}\) | Definition of Pointwise Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {f \odot g} } a \odot \map {\paren {f \odot g} } b\) | as $f \odot g$ is a homomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f a \odot \map g a} \odot \paren {\map f b \odot \map g b}\) | Definition of Pointwise Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {w \odot y} \odot \paren {x \odot z}\) | 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 $f: T \to S$ and $g: T \to S$ be arbitrary homomorphisms.
Then:
\(\ds \forall x, y \in T: \, \) | \(\ds \map {\paren {f \odot g} } {x \circledast y}\) | \(=\) | \(\ds \map f {x \circledast y} \odot \map g {x \circledast y}\) | Definition of Pointwise Operation | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \odot \map f y} \odot \paren {\map g x \odot \map g y}\) | as $f$ and $g$ are both homomorphisms | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \odot \map g x} \odot \paren {\map f y \odot \map g y}\) | Definition of Entropic Structure | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map {\paren {f \odot g} } x} \odot \paren {\map {\paren {f \odot g} } y}\) | Definition of Pointwise Operation |
Hence by definition $f \odot g$ 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 $13.13$