One-Parameter Semigroups in a Monoid

In the setting of Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(M,\cdot)$ be a monoid with identity $e$, products written by juxtaposition as in Semigroups (i).

(i) One-parameter semigroups. A one-parameter semigroup in $M$ is a function $T$ from the ray $[0,\infty)$ of Intervals of the Real Line (iii) to $M$ such that $T(0)=e$ and

$$T(s+t)=T(s)\,T(t)\qquad\text{for all } s,t\in[0,\infty).$$