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

Abstract

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 $H(A_q) = \left\{h \in \mathbf{N}: A^h = \bigcap_{q \in Q} A_q^h \right\}$ is investigated.