Problems and results on intersections of product sets and sumsets in semigroups

For every subset $A$ of a semigroup $S$, let $A^h$ be the set of all products of $h$ elements of $S$. If $(A{)}_{q\in Q}$ is a family of subsets of $S$, then $A={\bigcap }_{q\in Q}{A}_q$ satisfies $A^h\subseteq {\bigcap }_{q\in Q}{A}_q^h$. The product intersection set $H(A_q)=\{h\in \mathbf{N}:A^h={\bigcap }_{q\in Q}{A}_q^h\}$ is investigated.