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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

  1. $(y * f)(x*x')$
  2. $y*f(x*x')$
  3. $y*[f(x)*f(x')]$
  4. $[y*f(x)]*[y*f(x')]$
  5. $(y*f)(x) * (y*f)(x')$

A couple of basic observations for interested parties.

Let $Y$ denote a magma.

  1. If $Y$ is medial and $y \in Y$ is idempotent, then $y$ has the property of interest. (Easy exercise).

  2. 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 $i$).

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tylerco816 seems to have answered your question, so I will just offer some food for thought. If you add in an extra condition that, for every $a,b\in Y$, $\exists!c\in Y$ such that $a*c=b$, then you would have a Rack, or a Quandle if you further add that, for all $a\in Y$, $a*a=a$. My topology professor has done some work in knot theory with quandles, which I find is very interesting. – Brian Scholl Dec 21 '13 at 10:45
@BrianScholl, thanks! They look totally cool, exactly the kind of thing I was looking for. – goblin Dec 21 '13 at 10:48
Ha! I just realized that my professor is even listed in the external links for the Wikipedia page. – Brian Scholl Dec 21 '13 at 10:56
@BrianScholl, what's his name? – goblin Dec 21 '13 at 11:13
J. Scott Carter. His article A Survey of Quandle Ideas is listed in the external links for the Racks and Quandles wikipedia page. – Brian Scholl Dec 21 '13 at 11:14
up vote 2 down vote accepted

Apparently they are called (left) distributive magmas

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