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

TL;DR

This paper investigates the structure and properties of intersections of product sets and sumsets within semigroups, focusing on the behavior of the set of exponents where these intersections coincide.

Contribution

It introduces new results on the relationships between product set intersections and sumsets in semigroups, expanding understanding of their algebraic structure.

Findings

Characterization of the set H(A_q) where product powers of intersections match intersections of powers.

Conditions under which the intersection of multiple product sets equals the product of their intersections.

New bounds and properties related to sumsets and product sets 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.