Let $X$ and $Y$ denote magmas, suppose $f$ is a homomorphism $X \rightarrow Y$, and let $y \in Y$ satisfy the following condition.
$$\forall a,b \in Y : y*(a*b) = (y*a)*(y*b)$$
Then $y * f$ is a homomorphism.
Question. Is there a name for magmas $Y$ in which every $y \in Y$ satisfies the aforementioned condition?
Proof of claim. The following are equal
- $(y * f)(x*x')$
- $(y*f)(x) * (y*f)(x')$
A couple of basic observations for interested parties.
Let $Y$ denote a magma.
If $Y$ is medial and $y \in Y$ is idempotent, then $y$ has the property of interest. (Easy exercise).
Conversely, if $y \in Y$ satisfies the property of interest condition, then we may deduce that $y$ is idempotent, so long as there exists an idempotent $i \in Y$ that is also a right-identity of $y$. (Proof. Just put $a$ and $b$ equal to $i$).