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:

Hence the result by definition of entropic structure.