# 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 $13.12 \ \text{(g)}$