Pith. sign in

REVIEW 3 major objections 2 minor 21 references

The automorphism group of the zero-divisor digraph of matrices over an antiring

T0 review · 3 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper determines the full automorphism group of the zero-divisor digraph of $M_n(S)$ for antinegative commutative semirings with finitely many zero-divisors: every automorphism is a permutation of diagonal blocks, a row/column…

desk verdict Lemma 3.2 is false, so the main classification theorem is unproven; the decomposition method is genuinely new but the proof needs repair. read the letter →

arxiv 1908.04614 v1 pith:CNC7IKFD submitted 2019-08-13 math.AC math.COmath.RA

David Dolžan , Gabriel Verret This is my paper · ORCID
classification math.ACmath.COmath.RA MSC 05C6016Y6005C25
keywords zero-divisordigraphautomorphismgroupsemiringantiringmatrixtwinverticeswreathproductzerosum-free
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 aims to describe all symmetries of the zero-divisor digraph of a matrix semiring $M_n(S)$ when $S$ is a commutative antiring—a zerosum-free semiring such as the nonnegative integers, a distributive lattice, or the Boolean semiring—with finitely many zero-divisors. It claims that every automorphism of this digraph has a rigid three-part form: it permutes the components of a maximal decomposition of a non-zero-divisor, applies a simultaneous row/column permutation inside each component, and applies an isomorphism between the corresponding component digraphs. If correct, this reduces the automorphism-group problem to the smaller problem of understanding indecomposable antirings and directly generalizes earlier descriptions of automorphism groups of zero-divisor graphs of matrix rings over finite fields. The result matters because zero-divisor graphs are used to expose algebraic structure through graph-theoretic invariants, and matrix semirings arise naturally in optimization and discrete-event systems.

What carries the argument

The load-bearing objects are twin vertices and the decomposition of a maximal non-zero-divisor. Two vertices are twins when they have identical in- and out-neighborhoods; automorphisms preserve the twin relation, so every automorphism induces a permutation of the twin quotient. Theorem 3.4 identifies twins in $\Gamma(M_n(S))$: two matrices are twins exactly when their row-tuples and column-tuples of annihilator intersections coincide. The proof's engine is Lemma 3.2, which claims that $\sigma(A+B)$ and $\sigma(A)+\sigma(B)$ are twin vertices and therefore equal; the paper uses this claim repeatedly to decompose matrices as $\alpha A=e_1A+\cdots+e_sA$ and to force the componentwise action that yields the wreath-product structure.

What would settle it

Work in $S=\mathbb{N}_0$ with the usual operations. In $\Gamma(\mathbb{N}_0)$ the only products equal to zero involve $0$, so every positive integer is a twin of every other positive integer. The map fixing $0$ and swapping $1$ and $2$ is therefore an automorphism, yet $\sigma(1+1)=\sigma(2)=1$ while $\sigma(1)+\sigma(1)=2+2=4$, contradicting the additivity-up-to-twins equality used in the proof of Theorem 3.12.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.12. Let $S$ be a commutative antiring with identity and let $\alpha\in S\setminus Z(S)$ be a non-zero-divisor of maximal length $s$, written $\alpha=e_1+\cdots+e_s$ with all $e_i\neq 0$ and $e_i e_j=0$ for $i\neq j$. Every automorphism $\sigma$ of $\Gamma(M_n(S))$ is obtained from a permutation $\omega$ of $\{1,\dots,s\}$, permutations $\pi_i$ of $\{1,\dots,n\}$, and digraph isomorphisms $\tau_i\colon \Gamma(e_iS)\to \Gamma(e_{\omega(i)}S)$ by $\sigma(A)=\sum_{i=1}^s(\theta_{\pi_i}\circ\tau_i)(e_iA)$, where $\theta_\pi$ permutes rows and columns simultaneously. Conversely, any such choice is an automorphism. The paper further packages the whole automorphism group as a semidirect product of the regular automorphisms with a wreath product of $\mathrm{Sym}(n)\times \mathrm{Aut}(\overline{\Gamma}(e_iS))$ over the isomorphism classes of the components.

Load-bearing premise

The proof needs automorphisms to send the sum of two matrices to the sum of their images whenever the two sides are twin vertices; in the zero-divisor digraph of the nonnegative integers this equality fails, so the premise is not generally true.

Editorial extensions

If this is right

  • For a semiring with finitely many zero-divisors, computing $\mathrm{Aut}(\Gamma(M_n(S)))$ reduces to knowing the isomorphism classes of the component digraphs $\Gamma(e_iS)$ and the sizes of their twin classes.
  • If $S$ is indecomposable, the theorem says automorphisms consist only of simultaneous row/column permutations composed with automorphisms of $\Gamma(S)$, together with regular automorphisms inside twin classes.
  • The explicit semidirect-and-wreath formula makes group-theoretic invariants of the zero-divisor digraph computable from the coefficient semiring $S$ itself.
  • The result extends the known automorphism-group descriptions for matrix rings over finite fields to a wider class of zerosum-free coefficient semirings.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The main question left open by the proof is whether the claimed componentwise formula survives under an extra rigidity condition on the twin quotient, for example requiring the quotient to admit no nontrivial automorphisms.
  • Checking the Boolean semiring and other concrete antirings directly would reveal whether the decomposition holds in the cases the theorem was designed for even though its proof's additivity step needs repair.
  • The maximal-length decomposition method suggests a route to a similar componentwise statement for semirings with well-founded annihilator structures but infinitely many zero-divisors.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The paper studies the zero-divisor digraph Gamma(M_n(S)) of the semiring of n by n matrices over a commutative antiring S with identity. The main result (Theorem 3.12) claims a complete description of Aut(Gamma(M_n(S))) in terms of a permutation of the components e_i of a maximal-length non-zero-divisor alpha, permutations of rows and columns, and isomorphisms between the zero-divisor digraphs of the components e_i S. The abstract specializes this to antirings with finitely many zero-divisors. The proof strategy is to use the characterization of twin vertices in Theorem 3.4 to reduce automorphisms to componentwise maps that preserve single-entry matrices.

Significance. If valid, the result would generalize a substantial line of work on automorphism groups of zero-divisor graphs of matrix rings over finite fields to a much broader class of semirings, and the proposed normal form is a natural and potentially useful description. The paper is clearly organized, and the reduction to indecomposable subsemirings is conceptually appealing. However, the proof rests on a false additivity lemma, and the counterexample occurs in a semiring that satisfies the hypotheses stated in the abstract, so the main theorem is not established.

major comments (3)
  1. [Section 3, Lemma 3.2] The lemma asserts that sigma(A+B) and sigma(A)+sigma(B) are twin vertices and therefore equal. The proof establishes only that the two elements have the same left and right annihilators, which is the definition of being twin vertices; equality does not follow. In the antiring N0 of nonnegative integers with the usual operations, every positive integer has N^+(x)=N^-(x)={0}, so all positive integers are twins. The permutation fixing 0 and transposing 1 and 2 is an automorphism of Gamma(N0), yet sigma(1+1)=sigma(2)=1 while sigma(1)+sigma(1)=2+2=4. Hence the lemma is false.
  2. [Lemmas 3.6, 3.8, 3.10, Corollary 3.11, Theorem 3.12] The equality in Lemma 3.2 is load-bearing throughout the componentwise decomposition: Lemma 3.6 uses it to rewrite e_r E_ij as a sum of sigma^{-1}(e_k B), Proposition 3.8 and Lemma 3.10 use it to compare sums of single-entry matrices, and the final line of Theorem 3.12 invokes it to pass from sigma(e_1 A + ... + e_s A) to the sum of the sigma(e_i A). Since graph automorphisms need not preserve addition, these steps are not justified. The counterexample to Lemma 3.2 occurs in S=N0, which is a commutative antiring with identity and finitely many zero-divisors, so the gap directly affects the claimed main result.
  3. [Theorem 3.12 proof] The displayed step 'By Theorem 3.4, A = alpha A = e_1 A + ... + e_s A' is not generally true unless alpha is the multiplicative identity. In N0, alpha=2 is a non-zero-divisor of maximal length, but A=1 is not equal to 2A. If the intended statement is that A and alpha A are twin vertices, then the subsequent conclusion would still require the false Lemma 3.2 in order to replace a twin relation by equality.
minor comments (2)
  1. [Remark 3.16] The phrase 'a sum of infinitely many mutually orthogonal zero-divisors' is not a well-formed expression in a semiring, since semiring addition is defined only for finite sums; the intended hypothesis should be formulated in terms of arbitrarily long finite decompositions or an explicitly completed infinite sum.
  2. [Throughout] There are several typographical and OCR-type errors in the title and abstract ('MA TRICES', 'dig raph', 'semi ring'); these should be corrected if the manuscript is revised.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, though the proof may contain a genuine mathematical flaw (a false lemma), which is a correctness issue rather than circular reasoning.

full rationale

The paper is a direct mathematical derivation from standard definitions: it defines the zero-divisor digraph Gamma(S) with vertex set S and edges u->v iff uv=0, then derives the automorphism group of Gamma(M_n(S)) through a sequence of lemmas (3.1, 3.2, 3.4, 3.6, 3.7, 3.9, 3.10, 3.11) leading to Theorem 3.12. There are no fitted constants, no subset of data being used to infer a related quantity, and no empirical prediction. The citations to prior work on zero-divisor graphs of rings, matrix rings, and semirings serve as background and comparison results, not as load-bearing justification for the target theorem. The proof is not circular: Theorem 3.12 is not assumed at any point, and no equation is defined in terms of the result it is supposed to establish. The reviewer's concern is that Lemma 3.2 asserts that if sigma(A+B) and sigma(A)+sigma(B) are twin vertices then they are equal, which may fail even though they have the same annihilators. For example, in Gamma(N0), the permutation fixing 0 and swapping 1 and 2 is an automorphism but sigma(1+1)=1 while sigma(1)+sigma(1)=4. This is a potential logical gap or false lemma, not circular reasoning: the proof may be invalid, but it is not self-referential in the sense of reducing to its own conclusions or fitting its inputs. Consequently, the circularity score is 0, and the substantive mathematical objection belongs under correctness risk rather than circularity analysis.

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

There are no fitted parameters and no invented entities. The only substantive inputs are the antiring hypotheses and the existence of a maximal decomposition. The proof additionally assumes, incorrectly, that graph automorphisms preserve addition for twin vertices.

assumptions (3)
  • domain assumption S is a commutative antinegative semiring with identity and a finite set of zero-divisors.
    This is the hypothesis of the main theorem and the abstract; it restricts the class of semirings considered.
  • domain assumption There exists alpha in S minus Z(S) of maximal length s with an orthogonal decomposition alpha = e1 + ... + es.
    This is needed to run the componentwise decomposition in Lemma 3.6 and Theorem 3.12. The paper asserts in Remark 3.16 that finiteness of Z(S) guarantees such a maximal decomposition exists.
  • standard math Standard background in semiring theory, graph theory, and automorphism groups.
    The proofs use standard definitions and elementary facts about graphs, semirings, and permutations without developing them.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The automorphism group of the zero-divisor digraph of matrices over an antiring." pith.science (2026). https://pith.science/paper/CNC7IKFD

@misc{pith2026190804614,
  author       = {Pith},
  title        = {Pith review of: The automorphism group of the zero-divisor digraph of matrices over an antiring},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CNC7IKFD}},
  note         = {Machine review of arXiv:1908.04614}
}
read the original abstract

We determine the automorphism group of the zero-divisor digraph of the semiring of matrices over an antinegative commutative semiring with a finite number of zero-divisors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    D. F. Anderson, P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447

    work page 1999

  2. [2]

    S. E. Atani, The zero-divisor graph with respect to ideal s of a commutative semiring, Glas. Mat. 43 (2008) 309–320

    work page 2008

  3. [3]

    S. E. Atani, An ideal-based zero-divisor graph of a commu tative semiring, Glas. Mat. 44 (2009), 141–153

    work page 2009

  4. [4]

    Baccelli, J

    F. Baccelli, J. Mairesse, Ergodic theorems for stochast ic operators and discrete event networks, Idem- potency, Cambridge University Press (1998), 171–208

    work page 1998

  5. [5]

    Beck, Coloring of commutative rings, J

    I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226

    work page 1988

  6. [6]

    Cannon, K

    A. Cannon, K. Neuerburg, S. P. Redmond, Zero-divisor gra phs of nearrings and semigroups, in Near- rings and Nearfields , eds. H. Kiechle, A. Kreuzer, M.J. Thomsen (Springer, Dordr echt, The Nether- lands, 2005), 189–200

    work page 2005

  7. [7]

    R. A. Cuninghame-Green, Minimax algebra, volume 166, Sp ringer Science & Business Media, 2012

    work page 2012

  8. [8]

    F. R. DeMeyer, T. McKenzie, K. Schneider, The zero-divis or graph of a commutative semigroup, Semigroup Forum 65 (2002) 206–214

    work page 2002

Show all 21 references
  1. [9]

    Dolˇ zan, P

    D. Dolˇ zan, P. Oblak, The zero-divisor graphs of rings an d semirings. Internat. J. Algebra Comput. 22 (2012), 20 pp

    2012

  2. [10]

    Han, The zero-divisor graph under group actions in a n oncommutative ring, J

    J. Han, The zero-divisor graph under group actions in a n oncommutative ring, J. Korean Math. Soc. 45 (2008), 1647–1659

    2008

  3. [11]

    Hebisch, H

    U. Hebisch, H. J. Weinert, Semirings: algebraic theory and applications in computer s cience, Series in Algebra, vol. 5 (World Scientific Publishing Co., Inc., Rive r Edge, NJ, 1998.)

    1998

  4. [12]

    Li, A heuristic method to compute the approximate pos tinverses of a fuzzy matrix, IEEE Trans- actions on Fuzzy Systems , 22 (2014), 1347–1351

    P. Li, A heuristic method to compute the approximate pos tinverses of a fuzzy matrix, IEEE Trans- actions on Fuzzy Systems , 22 (2014), 1347–1351

    2014

  5. [13]

    (2019), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A048291

    OEIS Foundation Inc. (2019), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A048291

    2019

  6. [14]

    S. Park, J. Han, The group of graph automorphisms over a m atrix rings, J. Korean Math. Soc. 48 (2011), 301–309. AUTOMORPHISM GROUP OF ZERO-DIVISOR DIGRAPH 9

    2011

  7. [15]

    Patil, B

    A. Patil, B. N. Waphare, V. Joshi, Perfect zero-divisor graphs, Discrete Math. 340 (2017), 740–745

    2017

  8. [16]

    S. P. Redmond, The zero-divisor graph of a non-commutat ive ring, Int. J. Commut. Rings 1 (2002), (1999) 203–211

    2002

  9. [17]

    Wang, A note on automorphisms of the zero-divisor gra ph of upper triangular matrices, Linear Algebra Appl

    L. Wang, A note on automorphisms of the zero-divisor gra ph of upper triangular matrices, Linear Algebra Appl. 465 (2015), 214–220

    2015

  10. [18]

    Wang, Automorphisms of the zero-divisor graph of the ring of all n × n matrices over a finite field, Discrete Math

    L. Wang, Automorphisms of the zero-divisor graph of the ring of all n × n matrices over a finite field, Discrete Math. 339 (2016) 2036–2041

    2016

  11. [19]

    D. Wong, X. Ma, J. Zhou, The group of automorphism of a zer o-divisor graph based on rank one upper triangular matrices, Linear Algebra Appl. 460 (2014), 242–258

    2014

  12. [20]

    S. Zhao, X. Wang, Invertible matrices and semilinear sp aces over commutative semirings, Information Sciences 180 (2010), 5115–5124

    2010

  13. [21]

    J. Zhou, D. Wong, X. Ma, Automorphisms of the zero-divis or graph of the full matrix ring, Linear and Multilinear Algebra 65 (2017), 991–1002. F aculty of Mathematics and Physics, University of Ljubljan a, Jadranska 21, 1000 Lju- bljana, Slovenia IMFM, Jadranska 19, 1000 Ljubl...

    2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.