Semi-group in quantum open systems

Question

In the literature of Open Quantum System, one often comes across the following ($t_2>t_1,>0$):

Semi-group property of a map: $A(t_1+t_2,0) = A(t_2,0) A(t_1,0)$.

What does this mean physically, and why the name semi-group?

Answer 1

We have evolution equation :

$ \frac{dp_t}{dt}=Lp_t $

The solution is :

$ A_t=e^{tL} $

That satisfy composition law:

$ A_{t+s}=A_tA_s $