# Equivalent Conditions for Entropic Structure

## Theorem

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

This page gathers together some conditions on $\struct {S, \odot}$ which are logical equivalences to $\struct {S, \odot}$ being an entropic structure.

### Pointwise Operation is Homomorphism

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.

### Pointwise Operation of Homomorphisms from External Direct Product is Homomorphism

Let $\struct {S \times S, \otimes}$ denote the external direct product of $\struct {S, \odot}$ with itself:

- $\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in S \times S: \tuple {x_1, y_1} \otimes \tuple {x_2, y_2} = \tuple {x_1 \odot x_2, y_1 \odot y_2}$

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

Let $f \odot g$ denote the pointwise operation on $S^{S \times S}$ 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.

### Mapping from External Direct Product is Homomorphism

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.

## 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 $13.13$