Epimorphism Preserves Distributivity

Theorem
Let $$\left({R_1, +_1, \circ_1}\right)$$ and $$\left({R_2, +_2, \circ_2}\right)$$ be algebraic structures.

Let $$\phi: R_1 \to R_2$$ be an epimorphism.


 * If $$\circ_1$$ is left distributive over $$+_1$$, then $$\circ_2$$ is left distributive over $$+_2$$.
 * If $$\circ_1$$ is right distributive over $$+_1$$, then $$\circ_2$$ is right distributive over $$+_2$$.

Consequently, if $$\circ_1$$ is distributive over $$+_1$$, then $$\circ_2$$ is distributive over $$+_2$$.

That is, epimorphism preserves distributivity.

Proof
Throughout the following, we assume the morphism property holds for $$\phi$$ for both operations.

Left Distributivity
Suppose $$\circ_1$$ is left distributive over $$+_1$$. Then:

$$ $$ $$ $$ $$

So $$\circ_2$$ is left distributive over $$+_2$$.

Right Distributivity
Suppose $$\circ_1$$ is right distributive over $$+_1$$. Then:

$$ $$ $$ $$ $$

So $$\circ_2$$ is right distributive over $$+_2$$.

Distributive
If $$\circ_1$$ is distributive over $$+_1$$, then it is both right and left distributive over $$+_1$$.

Hence from the above, $$\circ_2$$ is both right and left distributive over $$+_2$$.

That is, $$\circ_2$$ is distributive over $$+_2$$.