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Factorizations in rational monogenic semidomains

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Pith's one-line read Rational monogenic semidomains S_q have unique factorization if and only if q is a positive integer or a unit fraction, and the same list of q also governs half-factoriality, the Krull property, and root-closure.

desk verdict Clean classification of the classical factorization hierarchy for the natural family S_q = N_0[q], with a useful new tool (the technical monoid T_q) and honest open atomicity. read the letter →

arxiv 2607.10178 v1 pith:YYBXSR2T submitted 2026-07-11 math.AC

Anna Deng , Felix Gotti , Jason Zeng This is my paper · ORCID
classification math.AC MSC 16Y6013F1513A0511R0913G05
keywords monogenicsemidomainrationaltechnicalfractionsuniquefactorizationhalf-factorialfiniteboundedKrull
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper begins a systematic study of factorization inside the monogenic semidomains S_q = N_0[q] generated by a positive rational q. It proves that unique factorization, half-factoriality, the Krull property, and root-closure of the multiplicative monoid are all equivalent, and hold precisely when q lies in the positive integers or their reciprocals. Finite factorization is completely classified by three elementary conditions on q (q at least 1, unit fraction, or denominator a prime power), while bounded factorization is shown equivalent to the ascending chain condition on principal ideals. The monoid of technical fractions T_q, a divisor-closed submonoid of the nonzero elements, is introduced as the object that encodes the decisive arithmetic. A reader who cares about extending classical factorization theory beyond rings will see that even the simplest nontrivial positive extensions of N_0 already force many of the familiar implications to collapse or reverse.

What carries the argument

The monoid of technical fractions T_q, the divisor-closed submonoid of S_q^* consisting of those nonzero elements whose prime factors (numerator and denominator) all divide the denominator of q; its infimum away from 1 decides the bounded-factorization and ACCP properties, and the same monoid supplies the density arguments that control finite factorization.

What would settle it

For a concrete q outside N union N^{-1} (for example q = 5/6 or 2/3), evaluate the explicit polynomial R(x) = x^8 + 2x^7 + 2x^6 + 2x^5 + x^4 + x^3 + x^2 + x + 1 at q and check whether the two claimed factorizations of R(q) into length 2 and length 3 really consist of atoms; if either factor is composite the UF-equals-HF equivalence fails.

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Extended reading notes

Core claim

Over the class of rational monogenic semidomains S_q (q positive rational), the following are equivalent: S_q is a unique factorization semidomain, S_q is half-factorial, S_q is Krull, and the multiplicative monoid of S_q is root-closed; all four hold if and only if q belongs to N union N^{-1}. Independently, S_q has the finite factorization property if and only if q is at least 1, or q is a unit fraction, or the denominator of q is a prime power; and the bounded factorization property is equivalent to the ascending chain condition on principal ideals (and both hold precisely when the technical fractions stay bounded away from 1).

Load-bearing premise

The claim that an irreducible polynomial over the nonnegative integers which is the unique representation of its value at q must be an atom of S_q rests on the uniqueness of optimal polynomial representations produced by the mass-moving reduction process that replaces multiples of n or d by lower- or higher-degree terms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper initiates a systematic study of factorization in the monogenic semidomains S_q = N_0[q] for q positive rational. It introduces the divisor-closed monoid T_q of technical fractions (elements of S_q^* whose prime support lies only on the primes dividing d(q)) and uses it, together with optimal polynomial representations and Frobenius numbers of associated numerical monoids, to characterize classical factorization properties. The main results are: S_q is a UFS if and only if it is an HFS if and only if it is Krull if and only if S_q^* is root-closed, and these hold precisely when q belongs to N union N^{-1}; S_q is an FFS if and only if q >= 1 or q is a unit fraction or d(q) is a prime power; and, when q is not a unit fraction, BF is equivalent to ACCP and both hold if and only if inf(T_q \ {1}) > 1. Atomicity of S_q for general q is left open.

Significance. The work supplies the first detailed factorization-theoretic picture of the simplest nontrivial monogenic extensions of N_0. The equivalences UF = HF = Krull = root-closed = (q in N union N^{-1}) and the clean FF and BF characterizations are new and sharp; they also give a positive instance of the Bi-UF Positive Conjecture inside the monogenic class. The technical monoid T_q is a useful new tool that organizes the arithmetic. Proofs are elementary, self-contained, and carefully written; the reduction process for optimal polynomials, the density argument for non-prime-power denominators, and the explicit length-2/length-3 factorizations of a fixed 0-1 polynomial are fully detailed and appear correct. Leaving atomicity open is honest and does not diminish the value of the equivalences that are proved.

minor comments (4)
  1. Lemma 5.1 has an incomplete statement ("let S_q be the rational monogenic semidomain semidomain generated by q.") and no explicit claim; the proof that follows is clear, but the lemma header should be repaired.
  2. Figure 1 and Figure 4 both display implication diagrams; a single consolidated diagram (or a clearer cross-reference) would reduce redundancy.
  3. A few typographical slips remain (e.g., "semidomain semidomain", "strenght", "paramter", "containt polynomial"). A final proofreading pass would clean them.
  4. In the proof of Theorem 5.2 the claim that an irreducible unique-representation polynomial yields an atom is correct, but a one-sentence reminder that the uniqueness is supplied by Corollary 3.8 for the 0-1 polynomials actually used would make the argument even more self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; all central equivalences are proved by direct first-principles arguments inside S_q^*, with prior self-citations used only as independent black-box lemmas on additive monoids or general FFMs.

full rationale

The paper is a pure algebraic factorization study. The load-bearing claims (UF iff HF iff q in N union N^{-1}; FF characterization via the three cases on q; BF iff ACCP iff inf(T_q \ {1}) > 1; root-closed iff Krull iff UF) are established by explicit constructions (optimal polynomial representations via the n/d reduction process, concrete length-2/3 factorizations of R(q) using 0-1 polynomials, density of technical fractions when d(q) has two primes, ascending chains built from products of technical fractions near 1, height arguments for root-closure). These do not reduce to their own inputs by definition or by fitting. Self-citations (e.g., additive atomicity of M_q from [8,17], increasing monoids are FFMs from [16]) supply independent lemmas whose statements do not include the target multiplicative equivalences; they are not uniqueness theorems forbidding alternatives inside the present proofs, nor ansatzes smuggled in. Atomicity of S_q is left open, consistent with non-circularity. No free parameters, no data fits, no renaming of known patterns as new predictions. Minor self-citation presence yields score 1 rather than 0; nothing higher is warranted.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

The paper works entirely inside standard commutative monoid and semiring theory. No free parameters are fitted. The only non-standard objects are the technical monoid T_q and the height function, both defined explicitly and used as tools rather than postulated entities. Background facts (numerical monoids, Frobenius numbers of geometric sequences, density of log-ratio sets) are standard or cited.

assumptions (3)
  • standard math Every monoid satisfying ACCP is atomic; every UFM is Krull and root-closed; every BFM satisfies ACCP.
    Used throughout Sections 2 and 5 as background factorization theory (Anderson-Anderson-Zafrullah, Halter-Koch).
  • domain assumption The additive monoid M_q is atomic iff q^{-1} not in N_{>=2}, and then ACCP/BF/FF hold for M_q precisely when q>=1.
    Theorem 2.2, cited from Gotti-Gotti and Correa-Morris-Gotti; used only for the bi-HF corollary, not for the multiplicative claims.
  • domain assumption Membership criterion: c/d(q)^k > B_{q,k} (or >= B_{q,infty}) implies c/d(q)^k in S_q.
    Lemma 3.3 / CrowdMath 2024, restated and proved; used for density arguments and root-closure.
invented entities (2)
  • Monoid of technical fractions T_q = omega_q^{-1}(0) independent evidence
    purpose: Divisor-closed submonoid that encodes the arithmetic of S_q^*; its infimum away from 1 decides BF/ACCP and supplies the dense divisors that destroy FF.
    Defined in (4.3)-(4.4) and used as the central technical tool; fully constructive, no independent physical existence claimed.
  • Height function h(r) on S_q independent evidence
    purpose: Measures the minimal power of d(q) needed to clear the denominator; used to prove non-root-closedness when q not integer or unit fraction.
    Defined in Section 6.1 via the nested numerical monoids M_{q,k}; purely combinatorial.

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Pith. "Pith review of Factorizations in rational monogenic semidomains." pith.science (2026). https://pith.science/paper/YYBXSR2T

@misc{pith2026260710178,
  author       = {Pith},
  title        = {Pith review of: Factorizations in rational monogenic semidomains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYBXSR2T}},
  note         = {Machine review of arXiv:2607.10178}
}
abstract

For $\alpha \in \mathbb{C}$, the monogenic semidomain generated by $\alpha$ is the smallest subsemiring $S_\alpha$ of the complex field $\mathbb{C}$ containing $\alpha$. We initiate a systematic study of the arithmetic and factorizations of the monogenic semidomains $S_q$ generated by rational parameters $q$. After some preliminaries, we introduce and investigate the monoid of technical fractions $T_q$, which is a divisor-closed submonoid of the multiplicative monoid of $S_q$ that encodes a significant amount of arithmetic information about $S_q$. We then study several fundamental factorization properties of $S_q$: the bounded factorization (BF) and finite factorization (FF) properties, the unique factorization (UF) property, and the half-factorial (HF) property. First, we prove that $S_q$ satisfies the UF property if and only if it satisfies the HF property, which happens when $q \in \mathbb{N} \cup \mathbb{N}^{-1}$. We determine all the positive rational values of the parameter $q$ for which $S_q$ satisfies the FF property. Then we show that, over the class of rational monogenic semidomains, the BF property is equivalent to the ascending chain condition on principal ideals. Finally, we prove that $S_q$ is a Krull semidomain if and only if it is root-closed, which happens precisely when $S_q$ satisfies the UF property.

Figures

Figures reproduced from arXiv: 2607.10178 by the authors.

Figure 1
Figure 1. The diagram shows the known implications among the atomic properties we consider in this paper in the general class of atomic semidomains. The diagram also emphasizes (with red marked arrows) that none of the shown implications is reversible in the class of semidomains. We conclude this section by describing the structure of this paper. In Section 2, we introduce the notation and terminology used throughout the pape… view at source ↗
Figure 2
Figure 2. The implications in the diagram above hold on the class of rational mono￾genic semidomains: they are the most relevant results established in this paper [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Every monoid with the UF property is a Krull monoid, while every Krull monoid has the FF property. Given a factorization z ∈ Z(M), we refer to the length of z as the number of atoms of Mred that appear in z (counting repetitions), and we let |z| denote the length of z: if z = a1 · · · aℓ for some a1, . . . , aℓ ∈ A (Mred) then ℓ is the length of z. For each m ∈ M, the set of lengths of m is defined as follows: L(m) … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: The diagram shows the strenght of the factorization and ideal-theoretical properties we study in this section. Also, the congruences above imply that n k | p iαk 1 − p jβk 2 . Therefore uk := ϵk q k = d k nk (rk − 1) = d k nk p iαk 1 p jβk 2 − 1  = d k [PITH_FULL_IM…

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