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arxiv: 2606.25227 · v1 · pith:UH5CCF56new · submitted 2026-06-23 · 🧮 math.AC

On near atomicity and a characterization of the FF property

Jonathan Du , Felix Gotti , Leo Hong This is my paper

Pith reviewed 2026-06-25 21:15 UTC · model grok-4.3

classification 🧮 math.AC
keywords near atomicityatomicityintegral domainfinite factorization domainIDF propertyfactorization
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The pith

An integral domain can be nearly atomic without being atomic, and finite factorization domains are exactly the nearly atomic IDF domains.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper answers an open question by constructing a nearly atomic integral domain that is not atomic. It strengthens an earlier characterization by showing that an integral domain has the finite factorization property precisely when it is nearly atomic and satisfies the IDF property. The authors also demonstrate that this characterization does not hold if near atomicity is replaced by almost atomicity, even when restricting to IDF domains. A sympathetic reader would care because these results clarify the relationships among different weakenings of atomicity in the context of factorization in integral domains.

Core claim

We construct an explicit nearly atomic integral domain that is not atomic. We prove that an integral domain is an FFD if and only if it is nearly atomic and IDF. Near atomicity cannot be weakened to almost atomicity in this characterization, even within the class of IDF domains.

What carries the argument

The explicit construction of a nearly atomic non-atomic integral domain together with the proof of the equivalence between FFD and (nearly atomic and IDF).

If this is right

  • An integral domain that is nearly atomic but not atomic exists.
  • The finite factorization property is equivalent to being nearly atomic and IDF.
  • The equivalence fails when near atomicity is replaced by almost atomicity for IDF domains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction provides a concrete test case for studying other properties related to factorization in domains.
  • Similar examples might exist or be constructible in related monoid settings beyond integral domains.

Load-bearing premise

The constructed integral domain in the paper satisfies near atomicity but fails to be atomic.

What would settle it

A check showing that the constructed domain does not meet the definition of near atomicity, or that there exists a nearly atomic IDF domain that is not an FFD.

Figures

Figures reproduced from arXiv: 2606.25227 by Felix Gotti, Jonathan Du, Leo Hong.

Figure 1
Figure 1. Figure 1: Atomicity (A) Near Atomicity (NA) Almost Atomicity (AA) / / [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The diagram shows the implications obtained by combining the IDF prop￾erty with atomicity, near atomicity, and almost atomicity. The first marked reverse implication (in red) is one of the two main results of this paper, while the second marked reverse implication fails in general. 2. Background 2.1. General Notation. Let Z, Q, and R be the sets of integers, rational numbers, and real numbers, respectively… view at source ↗
read the original abstract

A commutative cancellative monoid is atomic if every nonunit factors into atoms, and an integral domain is atomic if its multiplicative monoid of nonzero elements is atomic. Several weakenings of atomicity have been introduced and studied during the past decade, including near atomicity, almost atomicity, and quasi-atomicity. Although nearly atomic monoids that are not atomic were already known, whether there exist nearly atomic integral domains that are not atomic had remained open. We answer this question affirmatively by constructing an explicit nearly atomic integral domain that is not atomic. We also strengthen the classical Anderson--Anderson--Zafrullah characterization of the finite factorization property by proving that an integral domain is an FFD if and only if it is both nearly atomic and IDF. We conclude by showing that near atomicity cannot be weakened to almost atomicity in this characterization, even within the class of IDF domains.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper constructs an explicit nearly atomic integral domain that is not atomic, resolving an open question in the literature on weakenings of atomicity. It also proves that an integral domain is an FFD if and only if it is nearly atomic and an IDF, strengthening the Anderson--Anderson--Zafrullah characterization, and shows that almost atomicity cannot substitute for near atomicity in this equivalence even among IDF domains.

Significance. The explicit construction provides the first known example of a nearly atomic non-atomic integral domain, a concrete strength that allows direct verification of the properties. The strengthened iff characterization of FFDs is a clean and useful refinement of prior work. These results advance factorization theory by clarifying the relationships among near atomicity, IDF, and finite factorization properties.

minor comments (2)
  1. [§2] §2, Definition 2.3: the notation for near-atoms could be clarified by explicitly distinguishing the 'near' factorization length from standard atomic length to avoid reader confusion with almost atomicity introduced later.
  2. [Theorem 4.3] Theorem 4.3: the proof that almost atomicity fails to characterize FFDs among IDF domains would benefit from a brief remark on why the counterexample domain satisfies IDF but not the finite factorization property.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity; explicit construction and proof are independent of inputs

full rationale

The paper constructs an explicit nearly atomic non-atomic integral domain and proves an iff characterization of FFDs as nearly atomic + IDF domains. These steps rest on external prior definitions of atomicity, near atomicity, IDF, and FFD (from Anderson--Anderson--Zafrullah and related literature) rather than self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The existence claim is verified by direct construction satisfying the stated axioms, and the necessity direction follows from standard monoid/domain arguments without reducing to the paper's own inputs by construction. No enumerated circularity pattern applies.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Claims rest on standard definitions of atomicity, near atomicity, almost atomicity, IDF, and FFD drawn from prior literature (Anderson--Anderson--Zafrullah and related papers); no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard axioms and definitions of commutative cancellative monoids and integral domains
    The paper works inside established algebraic structures.

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discussion (0)

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Reference graph

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