Exponential functions on semigroups

Summary

An exponential function on a semigroup $S$ is a complex-valued function $m$ with $m(xy) = m(x) m(y)$ for all $x, y \in S$. On the additive group $\R$, the continuous ones are the functions $x \mapsto e^{\lambda x}$ with $\lambda \in \C$, and the zero function. Exponentials are the multiplicative solutions of Cauchy's equation on $S$, and they generalize the characters of abelian groups. Many functional equations on groups and semigroups have solutions built from a few exponentials, such as $\frac{1}{2}(m(x) + m(x^{-1}))$ for d'Alembert's equation.