Equivalent Conditions for Entropic Structure/Pointwise Operation is Homomorphism

Theorem
Let $\struct {S, \odot}$ be an algebraic structure.

Let $\struct {T, \circledast}$ be an arbitrary algebraic structure.

Let $f$ and $g$ be mappings from $\struct {T, \circledast}$ to $\struct {S, \odot}$.

Let $f \odot g$ denote the pointwise operation on $S^T$ induced by $\odot$.

Then:
 * If $f$ and $g$ are homomorphisms, then $f \odot g$ is also a homomorphism


 * $\struct {S, \odot}$ is an entropic structure.
 * $\struct {S, \odot}$ is an entropic structure.

Sufficient Condition
Let $\struct {S, \odot}$ be such that if $f$ and $g$ are homomorphisms, then $f \odot g$ is also a homomorphism.

So, let $f: T \to S$ and $g: T \to S$ be arbitrary homomorphisms.

Let $a, b \in T$ be arbitrary.

Because $T$ is arbitrary, and $f$ and $g$ are arbitrary, it follows that:

Thus we have:

As $w$, $x$, $y$ and $z$ are arbitrary, $\struct {S, \odot}$ is an entropic structure.

Necessary Condition
Let $\struct {S, \odot}$ be an entropic structure.

Let $f: T \to S$ and $g: T \to S$ be arbitrary homomorphisms.

Then:

Hence by definition $f \odot g$ is a homomorphism.