Intersections of sumsets in additive number theory

Let $A$ be a subset of an additive abelian semigroup $S$ and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q{)}_{q=1}^{\mathrm{\infty }}$ be a strictly decreasing sequence of sets in $S$ and let $A={\bigcap }_{q=1}^{\mathrm{\infty }}{A}_q$. When does one have $$hA=\bigcap _{q=1}^{\mathrm{\infty }}h{A}_q$$ for some or all $h\ge 2$?