Question on terminology If a map $M$ has the feature
$$
M(x+y)=M(y)M(x)
$$
is there a term to describe this kind of map? It's not a linear map, at least. The order matters here - $M(y)M(x) \neq M(x)M(y)$.
 A: Its name is an exponential map.
A: A map $M: A \to B$ with the property that
$$
\forall x, y \in A: M(x+y)=M(x)M(y) \tag{*}
$$
is called a group homomorphism from the (additive) group $(A, +)$ to the (multiplicative) group $(B, \cdot)$. Example:
$$
 M : \Bbb C \to \Bbb C,  z \mapsto e^{\operatorname{Re}(z)}.
$$
Your condition
$$
\forall x, y \in A: M(x+y)=M(y)M(x) \tag{**}
$$
is equivalent to $(*)$ if $(A, +)$ is commutative (i.e. $A$ is an abelian group), as mentioned in the comments. I am not aware of a term for $(**)$ in the non-commutative case.

In the particular case that $A=B= \Bbb R$, $(*)$ is equivalent to $f = \log M$ being an additive function on $\Bbb R$, i.e. it satisfies Cauchy's functional equation
$$ 
\forall x, y \in \Bbb R: f(x+y)=f(x) + f(y) \, .
$$
This functional equation has nonlinear solutions, but under some conditions (i.e. continuity) it follows that $f(x) = cx$ for some constant $c$. So in this special case we have that $M$ is an exponential function: $M(x) = e^{cx}$.
