I see how for an alebraic structure to be a group then we'd kind of need an inverse element, which is unnecessary for the solutions of DEs. But an identity element is pretty important, isn't it?
On the other hand on Wikipedia i found that an identity element
$$ T(0)=I $$
is defined, where $T$ is the map $\mathbb{R}_+ \to L(X)$ and $X$ is a Banach space.
So my question is why did we chose a semigroup for this purpose and not a monoid/group and why does it have an idenity element?