Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$.
Now for the main idea:
Definition 1. Let:
- $X,Y$ denote magmas, with $X$ surjective.
- $f,g : X \rightarrow Y$ denote functions, not necessarily homomorphisms.
Then there is at most one function $h : X \rightarrow Y$ that satisfies the following equation for all $x,x' \in X$. $$h(x*x') = f(x) * g(x')$$
If such a function exists, let us say that "$(f,g)$ has a homotope," and let us call it "the homotope of $(f,g).$"
This is my own terminology; it is inspired by the existing terminology "homotopy of magmas".
Main Question. I'm interested in sufficient conditions under which the homotope of $(f,g)$:
- exists, and/or
- not only exists, but also, happens to be a homomorphism.
The remainder of the question is discussion.
Proposition 0. Let $X$ and $Y$ denote any magmas, with $X$ surjective. Let $h : X \rightarrow Y$ denote a homomorphism. Then $(h,h)$ has a homotope, namely $h.$
The converse doesn't hold though:
Example. Let $X$ denote a set, and $\mathcal{P}(X)$ denote its powerset. Define a binary operation $\rightarrow$ on $\mathcal{P}(X)$ by writing $A \rightarrow B = A^c \cup B$. Organize $\mathcal{P}(X)$ as a magma, with binary operation $\rightarrow$. Then $\mathcal{P}(X)$ is surjective. Now given $A \subseteq X$, the function $(A \cap -) : \mathcal{P}(X) \rightarrow \mathcal{P}(X)$ that takes $B \subseteq X$ to $A \cap B \subseteq X$ is not, in general, a magma homomorphism. Nonetheless, the ordered pair of functions $(A \cap -, A \cap -)$ has a homotope, namely the function $(A \rightarrow -) : \mathcal{P}(X) \rightarrow \mathcal{P}(X)$ that takes $B \subseteq X$ to $A \rightarrow B \subseteq X$. Furthermore, the function $(A \rightarrow -)$ is always a magma homomorphism, for all $A \subseteq X$. Therefore, the converse to the above proposition is false.
Moving right along, the following theorem shows that if $X$ and $Y$ have identity elements, and if they're preserved by $f$ and $g$, then the concept "homotope" gives us nothing new beyond the concept "homomorphism."
Proposition 1. Suppose $X$ and $Y$ are magmas-with-identity. Then $X$ is surjective. Furthermore, if $f$ and $g$ are identity-preserving functions $X \rightarrow Y$ such that $(f,g)$ has a homotope, then $f$ equals $g,$ which equals the homotope of $(f,g)$, and they're all homomorphisms.
Secondary Question. Where can I get more information?