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 $\struct {S^S, \oplus}$.

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:

demonstrating that $f \oplus g$ is a homomorphism from $S$ to itself.

Hence the result by definition of closed algebraic structure.