Structure Induced on Set of Self-Maps on Entropic Structure is Entropic

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 $\odot$.

Then $\struct {S^S, \oplus}$ 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 $f, g, p, q \in S^S$ be arbitrary.

Let $x \in S$ be arbitrary.

Then:

\(\ds \map {\paren {\paren {f \oplus g} \oplus \paren {p \oplus q} } } x\)

\(=\)

\(\ds \paren {\map f x \odot \map g x} \odot \paren {\map p x \odot \map q x}\)

Definition of Pointwise Operation

\(\ds \)

\(=\)

\(\ds \paren {\map f x \odot \map p x} \odot \paren {\map g x \odot \map q x}\)

Definition of Entropic Structure

\(\ds \)

\(=\)

\(\ds \map {\paren {\paren {f \oplus p} \oplus \paren {g \oplus q} } } x\)

Definition of Pointwise Operation

\(\ds \leadsto \ \ \)

\(\ds \paren {f \oplus g} \oplus \paren {p \oplus q}\)

\(=\)

\(\ds \paren {f \oplus p} \oplus \paren {g \oplus q}\)

Equality of Mappings

Hence the result by definition of entropic 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 $13.12 \ \text{(g)}$