Why is $T(t+s)=T(t)T(s)$ called semigroup property? Let $\{T(t):t>0\}$ be a family of operators. We say it is semi group if
                   $$ T(t+s)=T(t)T(s) \text{ for all }t,s >0$$
On the other hand any set with an associative binary operation is called semi group.
I am unable to figured it out that why $T(t+s)=T(t)T(s)$ is called semi-group property.

I guess it's because the map $T:\mathbb R_+\to B(H)$ is a semigroup homomorphism (maps from one semigroup to another one, preserving the semigroup operation). In the same way that a representation $U:\mathbb R\to B(H)$ is a group homomorphism. 
