Global Product Intersection Sets in Semigroups

Abstract

For a family $(A_q{)}_{q\in Q}$ of subsets of a semigroup, the product intersection set records those exponents $h\in \mathbb{N}$ for which the $h$-fold product set of the intersection, $({\bigcap }_q{A}_q{)}^h$, is equal to ${\bigcap }_q{A}_q^h$, the intersection of the product sets. Nathanson recently asked which subsets of $\mathbb{N}$ can occur as a product intersection set, both for arbitrary and for decreasing families $(A_q{)}_{q\in Q}$. We solve both problems by giving a complete classification. In particular, when $|Q|\ge 2$, we show that in either case any subset $X\subseteq \mathbb{N}$ with $1\in X$ occurs as a product intersection set. Both classifications were autonomously discovered and formally verified in Lean by Aristotle, a formal reasoning agent developed by Harmonic.

8 pages