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arxiv: 2604.15498 · v1 · submitted 2026-04-16 · 🧮 math.CO

Complemented zero-divisor graph of posets

Anagha Khiste , Ganesh Tarte , Vinayak Joshi This is my paper

Pith reviewed 2026-05-10 10:18 UTC · model grok-4.3

classification 🧮 math.CO
keywords zero-divisor graphposetcomplemented graphquasi-complemented posetArtinian ringreduced semigroupvector space graph
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The pith

The zero-divisor graph of a poset with zero is complemented precisely when the poset is quasi-complemented.

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

The paper establishes equivalent conditions under which the zero-divisor graph of a poset with a zero element becomes a complemented graph. It characterizes these graphs using the notion of quasi-complemented posets. The authors also show that for any such poset, a complemented zero-divisor graph is the same as a uniquely complemented one. Algebraic and topological properties are used to provide further characterizations. These results are then applied to zero-divisor graphs in semigroups, comaximal graphs in rings, and union graphs in vector spaces.

Core claim

For a poset Q with zero, the zero-divisor graph Γ(Q) is complemented if and only if Q is quasi-complemented. Moreover, the complemented and uniquely complemented properties coincide for Γ(Q). Equivalent conditions are derived, along with algebraic and topological characterizations, which are applied to reduced semigroups with zero, Artinian rings, and finite vector spaces.

What carries the argument

The zero-divisor graph Γ(Q) on the nonzero elements of poset Q with 0, where adjacency holds when the meet is 0; complementation of this graph is equivalent to Q being quasi-complemented.

If this is right

  • The graph Γ(Q) is complemented exactly when Q is quasi-complemented.
  • Complemented and uniquely complemented versions of Γ(Q) are identical for any poset with zero.
  • Algebraic and topological characterizations determine when Γ(Q) is complemented.
  • The conditions extend to zero-divisor graphs of reduced semigroups, comaximal graphs of Artinian rings, and nonzero component union graphs of vector spaces.

Where Pith is reading between the lines

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

  • Graph complementation can serve as a proxy for verifying the quasi-complemented property directly from the order structure.
  • The link may transfer tools between order theory and graph theory for studying zero relations in other algebraic objects.
  • The coincidence of complemented and uniquely complemented cases could simplify uniqueness proofs in related graph constructions.

Load-bearing premise

The poset must have a zero element so that zero-divisors can be defined via pairs whose meet is zero.

What would settle it

A concrete poset Q with zero element whose zero-divisor graph is complemented but where Q fails to be quasi-complemented.

Figures

Figures reproduced from arXiv: 2604.15498 by Anagha Khiste, Ganesh Tarte, Vinayak Joshi.

Figure 1
Figure 1. Figure 1: A poset Q /∈ P ℓ MF P such that ({a, b} uℓ) ⊥ ̸= a ⊥ ∩ b ⊥ [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A quasicomplemented poset Q in P ℓ MF P without a.c. Let Q be a poset under set inclusion, where Q = {X ⊆ N| either X = {n} or X = N \ {n} for every n ∈ N}. The Hasse diagram of Q is shown in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

In this paper, we derive a set of equivalent conditions for the zero-divisor graph $\Gamma(Q)$ of a poset $Q$ with $0$ to be complemented, characterizing it in terms of quasi-complemented posets. Furthermore, we prove that the notions of a complemented zero-divisor graph and a uniquely complemented zero-divisor graph coincide for any poset $Q$ with $0$. In addition, we provide both algebraic and topological characterizations for $\Gamma(Q)$ to be a complemented graph. In the final section, we apply these characterizations to the zero-divisor graphs of a reduced (multiplicative) semigroup $S$ with $0$ and the comaximal (ideal) graph of an Artinian ring $R$, and the nonzero component union graph $\mathbb{UG}(\mathbb{V})$ of a finite-dimensional vector space $\mathbb{V}$ over a field $\mathbb{F}$.

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 / 4 minor

Summary. The paper derives a set of equivalent conditions for the zero-divisor graph Γ(Q) of a poset Q with zero element to be complemented, characterizing this property in terms of quasi-complemented posets. It further proves that the notions of complemented and uniquely complemented zero-divisor graphs coincide for any poset Q with 0. Algebraic and topological characterizations of when Γ(Q) is complemented are provided, with applications as corollaries to the zero-divisor graphs of reduced semigroups with 0, the comaximal ideal graph of Artinian rings, and the nonzero component union graph of finite-dimensional vector spaces over a field.

Significance. If the derivations hold, the work establishes a clean bridge between order-theoretic notions (quasi-complements in posets) and graph-theoretic complementation in zero-divisor graphs, with the coincidence result for complemented vs. uniquely complemented graphs being a notable simplification. The applications to semigroups, rings, and vector-space graphs demonstrate direct utility in combinatorial algebra. The characterizations are parameter-free and rest on standard definitions, which strengthens their applicability.

minor comments (4)
  1. [Section 3] In the statement of the main characterization theorem (likely Theorem 3.1 or equivalent), the definition of adjacency in Γ(Q) via a ∧ b = 0 should be restated explicitly for clarity, as it is used repeatedly in the equivalences.
  2. [Section 4] The topological characterization in the later section would benefit from a brief reminder of the topology in use (e.g., order topology or Zariski-type), since the transition from algebraic to topological conditions is not immediately obvious from the abstract.
  3. [Section 5] In the applications section, the corollary for the comaximal graph of an Artinian ring R should include a short note on why the poset of ideals satisfies the quasi-complemented hypothesis, to make the reduction self-contained.
  4. [Introduction and Section 2] A few instances of notation (e.g., use of Γ(Q) vs. Γ_0(Q)) appear inconsistently across the introduction and main theorems; uniformizing this would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its significance, and the recommendation of minor revision. No specific major comments or points requiring clarification were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes equivalent conditions linking complemented zero-divisor graphs Γ(Q) to quasi-complemented posets via standard order-theoretic definitions (a ∧ b = 0 for adjacency, quasi-complement via maximal join) and graph complementation. The coincidence of complemented and uniquely complemented graphs is shown directly from these definitions for any poset with zero. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear; the characterizations and corollaries to semigroups, rings, and vector spaces follow as independent theorems without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions from order theory and graph theory without introducing new free parameters or invented entities; the work uses existing notions of posets with zero, zero-divisor relations, and complemented graphs.

axioms (2)
  • domain assumption A poset Q has a least element 0, and the zero-divisor graph is defined by adjacency when the meet of two elements is 0.
    This is the foundational setup for Γ(Q) invoked throughout the characterizations.
  • standard math Standard definitions of complemented graphs and quasi-complemented posets from prior literature.
    Used to state the equivalent conditions and coincidence result.

pith-pipeline@v0.9.0 · 5452 in / 1401 out tokens · 34359 ms · 2026-05-10T10:18:33.804201+00:00 · methodology

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Reference graph

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