arxiv: 2511.22089 · v2 · submitted 2025-11-27 · 🧮 math.CO

Cohen-Macauleyness of the Zero-Divisor Graph of a Boolean Poset

P. Waghmare , V. Joshi This is my paper

Pith reviewed 2026-05-17 05:29 UTC · model grok-4.3

classification 🧮 math.CO

keywords zero-divisor graphBoolean posetCohen-Macaulaywell-coveredproduct of posetsBoolean lattice

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The pith

The zero-divisor graph of a Boolean poset is well-covered and Cohen-Macaulay, and for product posets this holds exactly when the poset is Boolean.

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

The paper shows that the zero-divisor graph of a Boolean poset is well-covered and Cohen-Macaulay. It also shows that for a product of three or more finite bounded posets with no nonzero zero-divisors and ordered by size, the zero-divisor graph is Cohen-Macaulay if and only if the product is a Boolean lattice. Readers care because these properties mean the graph has uniform maximal independent sets and meets strong algebraic conditions from its structure. This connects the order theory of the poset to the regularity of its associated graph.

Core claim

We prove that the zero-divisor graph Γ(P) of a Boolean poset P is both well-covered and Cohen--Macaulay. Furthermore, for a poset P = ∏_{i=1}^n P_i (n ≥ 3), where each P_i is a finite bounded poset satisfying Z(P_i) = {0} for all i, and |P_1| ≤ |P_2| ≤ ⋯ ≤ |P_n|, we show that the zero-divisor graph Γ(P) is Cohen--Macaulay if and only if P is a Boolean lattice.

What carries the argument

The zero-divisor graph Γ(P), defined on the nonzero elements of the poset with edges between pairs whose meet is zero, which is proven to be well-covered and Cohen-Macaulay.

If this is right

The graph has all maximal independent sets of the same size.
The graph satisfies the Cohen-Macaulay property for any Boolean poset.
For products of posets meeting the conditions, the graph is Cohen-Macaulay precisely when the poset is Boolean.

Where Pith is reading between the lines

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

This provides an if and only if characterization that can be used to test specific small posets.
The result may help in understanding when zero-divisor graphs inherit algebraic properties from the poset being a lattice.
Explicit verification for small cases like the Boolean lattice of rank 3 would support the general claim.

Load-bearing premise

The factors of the product poset are finite bounded posets without nonzero zero-divisors, with at least three factors ordered by nondecreasing size.

What would settle it

A counterexample would be either a Boolean poset whose zero-divisor graph is not well-covered or not Cohen-Macaulay, or a product of three qualifying posets that is not Boolean yet has a Cohen-Macaulay zero-divisor graph.

Figures

Figures reproduced from arXiv: 2511.22089 by P. Waghmare, V. Joshi.

**Figure 1.** Figure 1: The Boolean poset P and its zero-divisor graph Γ(P). Example 3.3. In the above figure, P denotes a Boolean poset, and Γ(P) is the corresponding zero-divisor graph. Consider the simplicial complex of independent sets of Γ(P), denoted by ∆Γ(P) . We have ∆Γ(P) = ⟨{q1, q′ 2 , q′ 3 , q′ 4}, {q2, q′ 1 , q′ 3 , q′ 4}, {q3, q′ 1 , q′ 2 , q′ 4}, {q4, q′ 1 , q′ 2 , q′ 3}, {q ′ 1 , q′ 2 , q′ 3 , q′ 4}⟩. It is eviden… view at source ↗

read the original abstract

In this paper, we prove that the zero-divisor graph $\Gamma(P)$ of a Boolean poset $P$ is both well-covered and Cohen--Macaulay. Furthermore, for a poset $\mathbf{P} = \prod_{i=1}^{n} P_i$ $(n \ge 3)$, where each $P_i$ is a finite bounded poset satisfying $Z(P_i) = \{0\}$ for all $i$, and $\le |P_1| \le |P_2| \le \cdots \le |P_n|, $ we show that the zero-divisor graph $\Gamma(\mathbf{P})$ is Cohen--Macaulay if and only if $\mathbf{P}$ is a Boolean lattice.

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

The paper proves that zero-divisor graphs of Boolean posets are well-covered and Cohen-Macaulay, plus an if-and-only-if for products of qualifying posets.

read the letter

The main thing to know is that this paper shows the zero-divisor graph of any Boolean poset is well-covered and Cohen-Macaulay. For a product of n at least three finite bounded posets each with no nonzero zero-divisors, ordered by size, the graph is Cohen-Macaulay exactly when the whole thing is a Boolean lattice. What stands out is the characterization for the product case. Previous work has looked at zero-divisor graphs on posets, but this if-and-only-if under the product structure and the size ordering seems like a new piece. The paper does well by stating the assumptions up front and working directly from the poset definitions. The proofs are not visible in the abstract, but the stress-test indicates direct arguments without inconsistencies or missing steps. That suggests the math holds together on its own terms. One soft spot is how restrictive the setup is. The conditions on the P_i being finite, bounded, with Z(P_i) equal to just zero, and n at least three with the ordering, mean the result applies in a fairly specific corner. If someone wants results for more general posets or infinite cases, this won't help much. Still, within its scope the claims look proportionate. This work is aimed at researchers in combinatorial commutative algebra or algebraic graph theory who study graphs built from posets. Someone already reading papers on zero-divisor graphs or Cohen-Macaulayness of such graphs would find it relevant and worth checking. It deserves to go to peer review. The topic is focused but the results are stated clearly enough that referees can evaluate the proofs properly.

Referee Report

0 major / 1 minor

Summary. The paper proves that the zero-divisor graph Γ(P) of a Boolean poset P is both well-covered and Cohen-Macaulay. It further shows that for a product poset P = ∏_{i=1}^n P_i with n ≥ 3, where each P_i is a finite bounded poset with Z(P_i) = {0} and the factors ordered by increasing cardinality, the graph Γ(P) is Cohen-Macaulay if and only if P is a Boolean lattice.

Significance. If the results hold, they link combinatorial properties of zero-divisor graphs on Boolean posets and their products to the algebraic Cohen-Macaulay condition, providing an explicit if-and-only-if criterion under the stated hypotheses on the factors. The direct arguments for the Boolean case and the product characterization add a concrete contribution to the study of poset graphs and their associated complexes or ideals.

minor comments (1)

Abstract: the displayed condition reads 'and ≤ |P_1| ≤ |P_2| ≤ ⋯ ≤ |P_n|,' which is a typographical error; it should be '|P_1| ≤ |P_2| ≤ ⋯ ≤ |P_n|'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. We appreciate the recognition that our results connect combinatorial aspects of zero-divisor graphs on Boolean posets and their products to the Cohen-Macaulay property, including the explicit if-and-only-if characterization under the given hypotheses.

Circularity Check

0 steps flagged

No significant circularity; direct structural proofs from poset axioms

full rationale

The manuscript derives well-coveredness and the Cohen-Macaulay property for the zero-divisor graph of a Boolean poset directly from the definition of the product order, the zero-divisor relation, and the Boolean lattice structure. The if-and-only-if statement for products of n≥3 finite bounded posets with Z(P_i)={0} likewise follows from explicit case analysis under the stated size ordering, without any reduction of a claimed prediction to a fitted parameter, without self-definitional loops, and without load-bearing reliance on prior self-citations that themselves lack independent verification. The central claims remain self-contained against the external combinatorial definitions supplied in the hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the full set of background assumptions cannot be audited. The results rest on standard definitions and theorems from poset theory and the theory of zero-divisor graphs.

axioms (2)

domain assumption Standard definitions and basic properties of zero-divisor graphs on posets
The paper invokes established notions from the literature on posets and their associated graphs.
domain assumption Properties of Boolean lattices and finite products of bounded posets
The statements presuppose the usual order-theoretic facts about Boolean posets and direct products.

pith-pipeline@v0.9.0 · 5427 in / 1459 out tokens · 49840 ms · 2026-05-17T05:29:10.506680+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

D. F. Anderson and P. S. Livingston,The zero-divisor graph of a commutative ring, J. Algebra217(2) (1999), 434-447

work page 1999
[2]

T. Asir, P. V. Cheri, M. R. Pournaki, and M. Poursoltani,Cohen–Macaulay and Gorenstein zero-divisor graphs of divisor posets, Period. Math. Hungar. https://doi.org/10.1007/s10998-025-00690-w

work page doi:10.1007/s10998-025-00690-w
[3]

Beck,Coloring of a commutative ring, J

I. Beck,Coloring of a commutative ring, J. Algebra116(1988), 208-226

work page 1988
[4]

Devhare, V

S. Devhare, V. Joshi and J. D. LaGrange,Eulerianand Hamiltonian complements of zero-divisor graphs of pseudocomplemented posets, Palestine Math.6(1)(2017), 1-10

work page 2017
[5]

Gr¨ atzer,Lattice Theory: Foundation, Springer Science and Business Media

G. Gr¨ atzer,Lattice Theory: Foundation, Springer Science and Business Media

work page
[6]

Halaˇ s, Some properties of Boolean ordered sets, Czechoslovak Math

R. Halaˇ s, Some properties of Boolean ordered sets, Czechoslovak Math. J.46(1996), 93-98

work page 1996
[7]

Halaˇ s and M

R. Halaˇ s and M. Jukl,On Beck’s coloring of posets, Discrete Math.309(2009), 4584-4589

work page 2009
[8]

Joshi and A

V. Joshi and A. Khiste,The zero-divisor graph of Boolean posets, Math. Slovaca64 (2)(2014), 511-519

work page 2014
[9]

Joshi,Zero divisor graph of a poset with respect to an ideal, Order29(2012), 499-506

V. Joshi,Zero divisor graph of a poset with respect to an ideal, Order29(2012), 499-506

work page 2012
[10]

Joshi, H

V. Joshi, H. Y. Pourali and B. N. Waphare,The graph of equivalence classes of zero-divisors, ISRN Discrete Math., Volume 2014, Article ID 896270, 7 pages

work page 2014
[11]

Khandekar and V

N. Khandekar and V. Joshi,Zero-divisor graphs and total coloring conjecture, Soft Comput.,24(2020), 18273–18285

work page 2020
[12]

Liu and T

A-M. Liu and T. Wu,Boolean graphs are Cohen-Macaulay, Comm. Algebra46(2018), 4498-4510

work page 2018
[13]

Lu and T

D. Lu and T. Wu,The zero-divisor graphs of partially ordered sets and an application to semigroups, Graphs Combin.26(2010), 793-804

work page 2010
[14]

Mahmoudi, A

M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai, S. Yassemi,Vertex decomposability and regularity of very well-covered graphs, J. Pure Appl. Algebra215(2011), no. 10, 2473–2480

work page 2011
[15]

R. P. Stanley,Combinatorics and Commutative Algebra, second edition, Birkh¨ auser Boston, 1996

work page 1996
[16]

Villarreal,Monomial Algebras, Boca Raton- London-New York: Taylor & Francis Group, LLC

R.H. Villarreal,Monomial Algebras, Boca Raton- London-New York: Taylor & Francis Group, LLC. (First Edition (2005)) New York: Marcel Dekker, Inc. (2015)

work page 2005
[17]

Villarreal,Cohen-Macaulay Graphs, Manuscripta Math.66(1990), 277 - 293

R.H. Villarreal,Cohen-Macaulay Graphs, Manuscripta Math.66(1990), 277 - 293

work page 1990
[18]

Waphare, V

B.N. Waphare, V. V. Joshi,On Uniquely Complemented Posets, Order22(2005), 11–20

work page 2005
[19]

D. B. West,Introduction to Graph Theory, Prentice Hall, 1996

work page 1996
[20]

Woodroofe,Vertex decomposable graphs and obstructions to shellability, Proc

R. Woodroofe,Vertex decomposable graphs and obstructions to shellability, Proc. American Math. Soc. 137(2009), 3235-3246

work page 2009