Why do we talk about $C_0$-semigroups and not $C_0$-monoids?

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?

• Just because it is named a semi-group, it doesn't mean it actually is a semi-group. Also, we care not that much about the semi-group, more about its action. Jan 1, 2021 at 18:17
• Logically, you are right. However, in the case of $[0,\infty)$ the name semi-group is quite appropriate because it is "half" of the additive group $\mathbb R$. There are other cases where the prefix semi is used in an instructive way, e.g., semi-continuity. But there are also cases where semi is used in a rather bad way, for example, semi-norms are almost norms with just one little detail missing (namely that $\|x\|=0$ implies $x=0$). Jan 3, 2021 at 11:54
• @SeverinSchraven It is a semi-group though. Any monoid is Jan 3, 2021 at 17:32
• @Jakobian I know that (in fact that was one of the things we were taught in the first weeks of university). The emphasis should be on the second part tough. The map the OP writes down is the action and not the semi-group itself. The real semi-group is $\mathbb{R}_+$, which is not really interesting to an analyst (but its action is). At the end of the day we want that the flow at time $t+s$ is the same as if we evolve first to time $t$ and then evolve for the time $s$. This boils down to be a semi-group action, but I am not aware of an instance where we really use any deep theory about monoids. Jan 3, 2021 at 19:11
• The reason we do not use a group is that in general flows are not given as an action of a group. Sometimes we cannot solve the equation backwards in time and so we are missing the negative half-axis. Sometimes we will not even get a semi-group action due to blow-up in finite time. Jan 3, 2021 at 19:18