Dynamical map of open quantum system Why it is said that the dynamical map of an open quantum system is a semi-group. I think the identity map is always an element. Hence it should be a monoid.
 A: It is a semigroup because it is irreversible, i.e. you cannot go "back in time". Mathematically speaking, it is probably correct to say it is a monoid as an algebraic structure; however in analysis it has been always called a semigroup of operators (in a banach space). It is, if you want, a representation of a monoid, i.e. a family of operators $(U(t))_{t\in\mathbb{R}_+}$ is called a semigroup if $U(0)=id$ and $U(t)U(s)=U(t+s)$ for any $t,s\in\mathbb{R}_+$.
Comment. The point, as Mark Mitchison said, is probably that the term semigroup better reflects the intuition of an irreversible process that is like a group, but only for positive times and it is not invertible. Saying that it is a monoid (i.e. something with less structure; every group is a monoid), it is like saying "we do not know if it is invertible or not, however it may be". But probably this is only a matter of taste and not so important; nevertheless historically the term semigroup has been widely used in analysis as well as in physics.
