Recognition: unknown
Global Product Intersection Sets in Semigroups
Pith reviewed 2026-05-10 03:36 UTC · model grok-4.3
The pith
In semigroups, any subset of natural numbers containing 1 can occur as the product intersection set for families of two or more subsets.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For a family of subsets of a semigroup the product intersection set is the collection of natural numbers h such that the h-fold product of the intersection of the family equals the intersection of the h-fold products of the members. The paper classifies all possible such collections for arbitrary families and for decreasing families. Specifically, for families with cardinality at least two, every subset of the natural numbers that contains the number 1 arises in this way.
What carries the argument
The product intersection set for a family of subsets, defined by the equality condition on products and intersections.
Load-bearing premise
The classification holds for arbitrary semigroups and the two specified types of families using standard definitions of products and intersections.
What would settle it
A semigroup and family of at least two subsets whose product intersection set either excludes 1 or fails to include some other specific h that the classification claims must be possible would disprove the result.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper resolves Nathanson's question on realizable product intersection sets in semigroups. For families (A_q)_{q in Q} of subsets, the product intersection set consists of those h in N where (intersection A_q)^h equals intersection (A_q^h). The authors give explicit classifications for both arbitrary and decreasing families, proving in particular that when |Q| >= 2 any X subset N with 1 in X arises in either case. The classifications were autonomously discovered and machine-checked in Lean.
Significance. The result supplies a complete answer to the question, with the Lean formalization providing machine-checked proofs of the semigroup constructions and the required equalities (intersection A_q)^h = intersection (A_q^h). This removes the usual risk of missed cases in combinatorial classifications and constitutes a clear strength of the work.
minor comments (1)
- The abstract states the main theorem but does not indicate the Lean verification; adding a sentence on the formalization would improve visibility of this contribution.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. We appreciate the recognition of the Lean formalization as a strength.
Circularity Check
No significant circularity; direct classification from definitions with independent Lean verification
full rationale
The paper derives a complete classification of product intersection sets directly from the standard semigroup definitions of product sets and intersections, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. For |Q| >= 2 it shows every X containing 1 is realizable in both arbitrary and decreasing cases. The Lean formalization by Aristotle provides machine-checked verification of the constructions and equalities, constituting independent external evidence rather than circular support. No step in the derivation chain collapses to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard axioms of set theory and the definition of a semigroup (associative binary operation).
Reference graph
Works this paper leans on
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[1]
Aristotle: IMO-level Automated Theorem Proving
T. Achim et al.,Aristotle: IMO-level Automated Theorem Proving, preprint, 2025.https://doi.org/10.48550/ arXiv.2510.01346
work page internal anchor Pith review arXiv 2025
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[2]
M. B. Nathanson,Problems and Results on Intersections of Product Sets and Sumsets in Semigroups, preprint, 2026.https://doi.org/10.48550/arXiv.2604.04781
work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.04781 2026
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[3]
M. B. Nathanson,Intersections of Sumsets in Additive Number Theory, preprint, 2026.https://doi.org/10. 48550/arXiv.2512.23574 Groningen, the Netherlands Email address:wonterman1@hotmail.com Harmonic, London, United Kingdom Email address:pietro.monticone@harmonic.fun School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P. R. China...
work page internal anchor Pith review arXiv 2026
discussion (0)
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