Equivalent Conditions for Entropic Structure/Mapping from External Direct Product is Homomorphism

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

Let $\struct {S \times S, \otimes}$ denote the external direct product of $\struct {S, \odot}$ with itself:
 * $\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \tuple {x_1, y_1} \otimes \tuple {x_2, y_2} = \tuple {x_1 \odot x_2, y_1 \odot y_2}$

Consider the operation $\odot$ as a mapping from $S \times S$ to $S$.

That is:
 * $\forall a, b \in S: \map \odot {a, b} = a \odot b$

Then:
 * $\odot: S \times S \to S$ is a homomorphism from $\struct {S \times S, \otimes}$ to $\struct {S, \odot}$


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

Sufficient Condition
Let $\struct {S, \odot}$ be such that $\odot: S \times S \to S$ is a homomorphism.

Let $\tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S$ be arbitrary.

We have:

and it is seen $\struct {S, \odot}$ is an entropic structure by definition.

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

Then:

Hence by definition $\odot: S \times S \to S$ is a homomorphism.