I have a question/conjecture about homomorphism of magmas. It is more of just an intuitive idea and is quite possibly entirely wrong, but I'm not sure of the best way to determine its validity.
If I have a possibly infinite set $S$ on which binary operations $\times$ and $+$ are defined, and a finite magma $(G, \otimes)$, does a surjective homomorphism of magmas $\phi: (S, \times)\to (G, \otimes)$ have to be periodic with respect to $+$? What if $\phi$ is not surjective? A proof or counterexample would be greatly appreciated!
EDIT: For this question, I'd be willing to consider answers that are close but not exact. If we need to put extra restrictions on $+$ or $\otimes$ or the properties of $\phi$ I'd like to see what can be done in those cases to!