Exponential functions on semigroups

Remarks

Example: on $S = (\R, +)$, $m(x) = e^{(1 + 2i) x}$ is an exponential, since $e^{(1 + 2i)(x + y)} = e^{(1 + 2i) x} e^{(1 + 2i) y}$. Non-example: on $S = (\R, +)$, $m(x) = 2 e^x$ is not an exponential, since $m(0 + 0) = 2$ while $m(0) m(0) = 4$. The zero function is an exponential; some authors exclude it. If $S$ is a group and $m$ is not identically zero, then $m(e) = 1$, $m$ never vanishes, and $m$ is a homomorphism into the multiplicative group $\C^{\times}$. An exponential on an abelian group whose values all have absolute value $1$ is a character of the group.