Prime ideals in the Boolean polynomial semiring

Kalina Mincheva; Naufil Sakran

arxiv: 2512.23839 · v2 · submitted 2025-12-29 · 🧮 math.AC · math.RA

Prime ideals in the Boolean polynomial semiring

Kalina Mincheva , Naufil Sakran This is my paper

Pith reviewed 2026-05-16 19:32 UTC · model grok-4.3

classification 🧮 math.AC math.RA

keywords prime idealsBoolean semiringpolynomial semiringideal classificationsemiring theoryBoolean semifield

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The pith

Prime ideals of the Boolean polynomial semiring B[x] form three classes indexed by integers.

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

This paper classifies every prime ideal in the one-variable polynomial semiring over the Boolean semifield. It partitions them into three families, each labeled by an integer. The work also shows that an earlier conjecture on the structure of these ideals is false. A reader would care because semirings appear in logic, optimization, and tropical mathematics, where knowing the primes determines the spectrum and radical behavior.

Core claim

We disprove a conjecture of F. Alarcón and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of B[x] into three classes, indexed by integers.

What carries the argument

The three integer-indexed families of prime ideals in B[x], obtained by analyzing elements under the Boolean semifield operations of max and min.

If this is right

Every prime ideal of B[x] can be described explicitly by its integer index.
The prime spectrum of B[x] is now fully known and can be used to study radicals and varieties over this semiring.
Questions about primary decomposition or associated primes in B[x] reduce to checking the three families.
Direct comparisons with prime ideals in other idempotent semirings become feasible.

Where Pith is reading between the lines

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

Similar integer-indexed groupings may appear when the same methods are applied to polynomials over other idempotent semirings.
The explicit list opens the door to algorithmic checks for primeness in B[x] for concrete coefficients.
Extensions to several variables would test whether the three-class pattern persists in higher dimensions.

Load-bearing premise

The Boolean semifield uses its standard max-or and min-and operations, and polynomials are formed in the usual way without extra relations that would collapse the ideal lattice.

What would settle it

Finding even one prime ideal of B[x] that lies outside all three described integer-indexed classes.

read the original abstract

In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers.

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.

Desk Editor's Note private letter to a colleague

This paper gives an explicit three-family classification of the prime ideals in B[x] and disproves the Alarcón–Anderson conjecture with direct case analysis.

read the letter

The main takeaway is that the authors replace the old conjecture with a clean partition of the primes in the Boolean polynomial semiring into three families indexed by integers. They construct each family explicitly, verify that every member is prime, and then show that any prime ideal must land in one of them by tracking constant terms and leading coefficients under the max/min operations. The argument stays elementary and avoids heavy machinery, which makes the disproof and the classification both easy to follow on the page. That is the real contribution: B[x] is the standard test object in this corner of semiring algebra, so having the spectrum written down in usable form is genuinely helpful for anyone who needs to compute or compare ideals later. The proof structure looks solid from the outline; the case division appears exhaustive and the constructions are concrete rather than abstract. One minor soft spot is that the precise definition of prime ideal in the semiring sense is not restated at the outset, and readers coming from different papers sometimes use slightly different conventions about zero-divisors or the quotient ring. A single sentence fixing the definition would remove any ambiguity without adding length. Otherwise the work is self-contained and the claims match the evidence presented. This paper is aimed at people already working in commutative algebra over semirings or in idempotent and tropical geometry. A specialist who needs the prime spectrum of B[x] for further calculations will find it immediately usable. It is not a broad reorganization of the field, but it settles a basic question cleanly. I would send it to peer review; the central claim is concrete enough that a referee can check it in reasonable time and the result is worth recording.

Referee Report

0 major / 3 minor

Summary. The paper disproves the Alarcón-Anderson conjecture and provides a complete classification of the prime ideals of the Boolean polynomial semiring B[x], grouping them into three families indexed by integers. The Boolean semifield uses the standard max/min operations, and the classification proceeds via explicit construction of the families together with a direct case analysis on constant terms and leading coefficients to show that every prime ideal belongs to one of the three classes.

Significance. The result is significant because it settles an open conjecture with an explicit, constructive classification of the entire prime spectrum of B[x]. The direct enumeration into three integer-indexed families, together with the verification that each constructed ideal is prime and that the case division is exhaustive, supplies a concrete description that can serve as a reference point for further work on prime ideals in semirings and related structures such as tropical varieties.

minor comments (3)

[Abstract] The abstract states the classification into three classes but does not name the indexing parameter or give a one-sentence description of each family; adding this would improve immediate readability.
[Introduction] In the introduction, the precise statement of the Alarcón-Anderson conjecture being disproved should be quoted verbatim before the disproof is announced.
[Preliminaries] Notation for the Boolean semifield (addition as max, multiplication as min) is standard but should be restated once in the preliminaries section for readers coming from ring theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the disproof of the Alarcón-Anderson conjecture and the explicit classification of prime ideals in B[x] into three integer-indexed families.

Circularity Check

0 steps flagged

Classification by direct case analysis with no reductions to inputs

full rationale

The paper classifies prime ideals of the Boolean polynomial semiring B[x] into three families indexed by integers via explicit construction and exhaustive case analysis on constant terms and leading coefficients. All steps rely on the standard max/min operations of the Boolean semifield and the free semiring structure on one generator; no parameters are fitted, no quantities are defined in terms of the classification itself, and the disproof of the external Alarcón-Anderson conjecture proceeds by direct counterexamples rather than self-referential closure. No self-citations are load-bearing for the central claim, and the argument remains self-contained against the usual definitions of semiring ideals and primeness.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on the standard definition of prime ideals in a commutative semiring and on the concrete multiplication table of the Boolean semifield; no free parameters or new entities are introduced.

axioms (2)

standard math The Boolean semifield B = {0,1} with 0+0=0, 0+1=1, 1+1=1 and multiplication 0·anything=0, 1·1=1 is a commutative semiring.
Invoked implicitly when the polynomial ring B[x] is formed.
standard math Prime ideals in a commutative semiring are defined by the usual absorption property: if ab lies in I then a or b lies in I.
Standard definition used throughout the classification.

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discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Alarcón, D

F. Alarcón, D. Anderson,Commutative semirings and their lattices of ideals, Houston Journal of Mathematics, Volume 20, No. 4, (1994)

work page 1994
[2]

A. Fink, J. Giansiracusa, N. Giansiracusa, J. MundingerProjective hypersurfaces in tropical scheme theory I: the Macaulay idealarXiv preprint arXiv:2405.16338 (2024)

work page arXiv 2024
[3]

Golan,The theory of semirings with applications in mathematics and theoretical computer science, Longman Sci & Tech., 54, (1992)

J. Golan,The theory of semirings with applications in mathematics and theoretical computer science, Longman Sci & Tech., 54, (1992)

work page 1992
[4]

D. Joó, K. Mincheva,Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials, Sel. Math. New Ser. (2017), doi:10.1007/s00029-017-0322-x

work page doi:10.1007/s00029-017-0322-x 2017
[5]

D. Joó, K. Mincheva,On the dimension of the polynomial and the Laurent polynomial semiring, J. of Algebra, vol. 507, p. 103–119 (2018)

work page 2018
[6]

D. Joó, K. Mincheva,Varieties of prime tropical ideals and the dimension of the coordinate semiringarXiv preprint arxiv:2501.18053 (2025)

work page arXiv 2025
[7]

Maclagan, F

D. Maclagan, F. Rincón,Tropical IdealsCompositio Mathematica. 154(3):640-670. doi:10.1112/S0010437X17008004 (2018) DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70115 Email address:kmincheva@tulane.edu DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70115 Email address:nsakran@tulane.edu

work page doi:10.1112/s0010437x17008004 2018

[1] [1]

Alarcón, D

F. Alarcón, D. Anderson,Commutative semirings and their lattices of ideals, Houston Journal of Mathematics, Volume 20, No. 4, (1994)

work page 1994

[2] [2]

A. Fink, J. Giansiracusa, N. Giansiracusa, J. MundingerProjective hypersurfaces in tropical scheme theory I: the Macaulay idealarXiv preprint arXiv:2405.16338 (2024)

work page arXiv 2024

[3] [3]

Golan,The theory of semirings with applications in mathematics and theoretical computer science, Longman Sci & Tech., 54, (1992)

J. Golan,The theory of semirings with applications in mathematics and theoretical computer science, Longman Sci & Tech., 54, (1992)

work page 1992

[4] [4]

D. Joó, K. Mincheva,Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials, Sel. Math. New Ser. (2017), doi:10.1007/s00029-017-0322-x

work page doi:10.1007/s00029-017-0322-x 2017

[5] [5]

D. Joó, K. Mincheva,On the dimension of the polynomial and the Laurent polynomial semiring, J. of Algebra, vol. 507, p. 103–119 (2018)

work page 2018

[6] [6]

D. Joó, K. Mincheva,Varieties of prime tropical ideals and the dimension of the coordinate semiringarXiv preprint arxiv:2501.18053 (2025)

work page arXiv 2025

[7] [7]

Maclagan, F

D. Maclagan, F. Rincón,Tropical IdealsCompositio Mathematica. 154(3):640-670. doi:10.1112/S0010437X17008004 (2018) DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70115 Email address:kmincheva@tulane.edu DEPARTMENT OFMATHEMATICS, TULANEUNIVERSITY, NEWORLEANS, LA 70115 Email address:nsakran@tulane.edu

work page doi:10.1112/s0010437x17008004 2018