Global Product Intersection Sets in Semigroups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908994919989248 |
|---|---|
| author | van Doorn, Wouter Monticone, Pietro Tang, Quanyu |
| author_facet | van Doorn, Wouter Monticone, Pietro Tang, Quanyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global Product Intersection Sets in Semigroups van Doorn, Wouter Monticone, Pietro Tang, Quanyu Combinatorics Group Theory Number Theory 11B13, 11B05, 11B75, 11P70, 22D99 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. |
| title | Global Product Intersection Sets in Semigroups |
| topic | Combinatorics Group Theory Number Theory 11B13, 11B05, 11B75, 11P70, 22D99 |
| url | https://arxiv.org/abs/2604.18869 |