Set of Endomorphisms on Entropic Structure is Closed in Induced Structure on Set of Self-Maps
Theorem
Let $\struct {S, \odot}$ be a magma.
Let $\struct {S, \odot}$ be an entropic structure.
Let $S^S$ be the set of all mappings from $S$ to itself.
Let $\struct {S^S, \oplus}$ denote the algebraic structure on $S^S$ induced by $\odot$.
Let $T \subseteq S^S$ denote the set of endomorphisms on $\struct {S, \odot}$.
Then $\struct {T, \oplus_T}$ is closed in $\struct {S^S, \oplus}$.
Converse does not hold
Let $T \subseteq S^S$ denote the set of endomorphisms on $\struct {S, \odot}$.
Let $\struct {T, \oplus_T}$ be closed in $\struct {S^S, \oplus}$.
Then it is not necessarily the case that $\struct {S, \odot}$ is an entropic structure.
Proof
Recall the definition of algebraic structure on $S^S$ induced by $\odot$:
Let $f: S \to S$ and $g: S \to S$ be self-maps on $S$, and thus elements of $S^S$.
The pointwise operation on $S^S$ induced by $\odot$ is defined as:
- $\forall x \in S: \map {\paren {f \oplus g} } x = \map f x \odot \map g x$
Let $f, g \in T$ be arbitrary.
That is, let $f: S \to S$, $g: S \to S$ be endomorphisms on $\struct {S, \odot}$.
Let $x, y \in S$ be arbitrary.
Then:
\(\ds \map {\paren {f \oplus g} } {x \odot y}\) | \(=\) | \(\ds \map f {x \odot y} \odot \map g {x \odot y}\) | Definition of Pointwise Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \odot \map f y} \odot \paren {\map g x \odot \map g y}\) | Definition of Endomorphism | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map f x \odot \map g x} \odot \paren {\map f y \odot \map g y}\) | Definition of Entropic Structure | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\map {\paren {f \oplus g} } x} \odot \paren {\map {\paren {f \oplus g} } y}\) | Definition of Pointwise Operation |
demonstrating that $f \oplus g$ is a homomorphism from $S$ to itself.
Hence the result by definition of closed algebraic structure.
$\blacksquare$
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 $13.12 \ \text{(g)}$