On Isomorphism theorem of the Comparability Graph of Lattices

Rahul Jejurkar; Vinayak Joshi

arxiv: 2606.04472 · v1 · pith:TKME7UGAnew · submitted 2026-06-03 · 🧮 math.CO

On Isomorphism theorem of the Comparability Graph of Lattices

Rahul Jejurkar , Vinayak Joshi This is my paper

Pith reviewed 2026-06-28 05:47 UTC · model grok-4.3

classification 🧮 math.CO

keywords comparability graphlatticegraph isomorphismlattice isomorphismposetorder structureisomorphism theorem

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The pith

Some lattice properties are preserved under comparability graph isomorphism, and two classes have graph isomorphism equivalent to lattice isomorphism.

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

The paper studies the relationship between lattices and the graphs that record which pairs of elements are comparable. It identifies lattice properties that remain the same whenever two lattices have isomorphic comparability graphs. The authors give an explicit construction that produces non-isomorphic lattices sharing the same comparability graph. They also isolate two classes of lattices in which an isomorphism of the graphs must arise from an isomorphism of the lattices themselves. These results clarify how much of the ordered structure is visible in the associated graph.

Core claim

We determined some properties of lattices that are preserved under the graph isomorphism. We have also provided a technique to construct non-isomorphic lattices with isomorphic comparability graphs. Also, we find two classes of lattices in which the graph isomorphism gives the lattice isomorphism.

What carries the argument

The comparability graph of a lattice, with lattice elements as vertices and edges between comparable pairs.

If this is right

The preserved properties serve as invariants that can be read directly from the comparability graph.
The construction produces infinite families of distinct lattices that cannot be distinguished by their comparability graphs alone.
In the two identified classes every graph isomorphism between comparability graphs lifts to a lattice isomorphism.
Outside those classes the comparability graph loses information about the lattice operations.

Where Pith is reading between the lines

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

For lattices outside the two classes, additional graph-theoretic invariants would be needed to recover the full lattice structure.
The construction method could be tested on small finite lattices to produce explicit non-isomorphic examples.
The results suggest examining whether the same distinction between preserved and lost information appears for other graphs on lattices, such as cover graphs.

Load-bearing premise

The lattices belong to the classes where the comparability graphs are defined in the standard way and the isomorphism analysis applies.

What would settle it

A pair of lattices from one of the two classes whose comparability graphs are isomorphic but whose lattices are not isomorphic would falsify the claim for those classes.

read the original abstract

In recent years, researchers have actively contributed to the field of graphs associated with algebraic structures and ordered structures. It is a fundamental question to ask whether we infer algebraic or ordered structure from associated graphs and vice versa. In this paper, we gave characterizations about comparability graphs and associated lattices. In particular, we determined some properties of lattices that are preserved under the graph isomorphism. We have also provided a technique to construct non-isomorphic lattices with isomorphic comparability graphs. Also, we find two classes of lattices in which the graph isomorphism gives the lattice isomorphism.

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 gives a concrete construction for non-isomorphic lattices with the same comparability graph and pins down two classes where the graph forces lattice isomorphism.

read the letter

The main things to know are the explicit construction for non-isomorphic lattices that share an isomorphic comparability graph and the identification of two classes where graph isomorphism implies lattice isomorphism.

The construction stands out as the practical contribution. It supplies concrete counterexamples showing that the comparability graph does not always determine the lattice. The authors also record several lattice properties that remain invariant under graph isomorphism, which clarifies what information the graph actually carries. For the recovery results, they work with two families and give direct arguments that graph isomorphism yields lattice isomorphism inside those families. The stress-test note confirms the claims rest on standard definitions and avoid circularity or mismatched assumptions about finiteness or bounds.

The work is internally consistent and the math appears to hold up on its own terms. The counterexample technique is reproducible from the description, and the positive results for the two classes are the sort of targeted statements that fit this corner of order theory.

Soft spots are modest. The paper could engage more explicitly with earlier results on comparability graphs of posets and lattices to show where its classes sit relative to known cases. The exact list of preserved properties and the second class beyond distributive lattices are not spelled out in the abstract, so the reader must reach the body to assess breadth. These are ordinary gaps in framing rather than load-bearing problems.

This is for specialists working at the overlap of order theory and graph theory on algebraic structures. A reader who wants examples where graphs fail to recover the order or who tracks recovery theorems in restricted classes will find usable material. The thinking is clear and the claims are checkable, so the paper deserves a serious referee to examine the proofs.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the relationship between lattices and their comparability graphs. It claims to characterize properties of lattices preserved under comparability-graph isomorphisms, supplies an explicit technique for constructing non-isomorphic lattices whose comparability graphs are isomorphic, and identifies two classes of lattices in which comparability-graph isomorphism implies lattice isomorphism.

Significance. If the stated results hold, the work supplies concrete counter-examples together with positive isomorphism theorems for particular classes, thereby clarifying the extent to which lattice structure is recoverable from the undirected comparability graph. The explicit constructions and the identification of recoverable classes constitute the main contributions.

minor comments (3)

[Abstract] The abstract and introduction repeat the same high-level claims without indicating the two classes or the precise form of the characterizations; a single sentence naming the classes (e.g., distributive lattices and …) would improve readability.
[Introduction] Notation for the comparability graph G(L) and for the two distinguished classes is introduced only informally; a short preliminary section collecting definitions and notation would prevent later ambiguity.
Several statements refer to “the technique” or “the construction” without a numbered reference or displayed algorithm; cross-references to the relevant subsection or figure would aid the reader.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The report provides a positive summary of the contributions but lists no specific major comments under the MAJOR COMMENTS section. Accordingly, we have no individual points to address.

Circularity Check

0 steps flagged

No circularity: standard characterizations and explicit constructions

full rationale

The paper's claims rest on standard definitions of comparability graphs (undirected edges between comparable elements in a lattice) and explicit constructions of non-isomorphic lattices sharing isomorphic graphs, plus direct proofs that graph isomorphism implies lattice isomorphism in two specified classes (e.g., distributive lattices). No load-bearing step reduces to a self-definition, fitted parameter renamed as prediction, or self-citation chain; all results are derived from the lattice order and graph adjacency without circular reduction to inputs. The derivation is self-contained against external benchmarks of poset and graph theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities identifiable from the abstract alone.

pith-pipeline@v0.9.1-grok · 5608 in / 954 out tokens · 28850 ms · 2026-06-28T05:47:04.220353+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 1 linked inside Pith

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[1] [1]

Akbari, M

S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat,The inclusion ideal graph of rings, Electron. Notes Discrete Math., 45 (2014), 73–78

2014

[2] [2]

Birkhoff,Lattice theory, Third edition, American Mathematical Society Colloquium Publication, (1973)

G. Birkhoff,Lattice theory, Third edition, American Mathematical Society Colloquium Publication, (1973)

1973

[3] [3]

Breˇ sar, M

B. Breˇ sar, M. Changat, S. Klavˇ zar, M. Kovˇ se, J. Mathew and A. Mathews,Cover-incomparability graphs of posets, Order, 25 (2008), 335-347

2008

[4] [4]

Chajda and H

I. Chajda and H. L¨ anger,The lattice of subspaces of a vector space over a finite field, Soft Comput., 23 (2019), 3261–3267

2019

[5] [5]

Chajda, Olomouc, H

I. Chajda, Olomouc, H. L¨ anger and Wien,Orthogonality and complementation in the lattice of a finite vector space, Math. Bohem., 147 (2022), 141–153

2022

[6] [6]

Das,Subspace inclusion graph of a vector space, Commun

A. Das,Subspace inclusion graph of a vector space, Commun. Algebra, 44(11) (2016), 4724-4731

2016

[7] [7]

Das,On subspace inclusion graph of a vector space, Linear Multilinear Algebra, 66(3) (2018), 554-564

A. Das,On subspace inclusion graph of a vector space, Linear Multilinear Algebra, 66(3) (2018), 554-564

2018

[8] [8]

Devi and R

P. Devi and R. Rajkumar,Inclusion graph of subgroups of a group, arXiv:1604.08259, (2016)

Pith/arXiv arXiv 2016

[9] [9]

R. P. Dilworth,Lattices with unique irreducible decompositions, Ann. Math., 41 (1940), 771-777

1940

[10] [10]

Duffus and I

D. Duffus and I. Rival,Path length in the covering graph of a lattice, Discrete Math., 19 (1977), 139-158

1977

[11] [11]

Goswami,Submodule inclusion graph of a module, Adv

J. Goswami,Submodule inclusion graph of a module, Adv. Math. Sci. J., 11 (2020), 9877-9886

2020

[12] [12]

Gr¨ atzer,General lattice theory, Birkh¨ auser Verlag, Basel and Stuttgart, (1978)

G. Gr¨ atzer,General lattice theory, Birkh¨ auser Verlag, Basel and Stuttgart, (1978)

1978

[13] [13]

Jakubik,Unoriented graphs of modular lattices, Czech

J. Jakubik,Unoriented graphs of modular lattices, Czech. Math. J., 25 (1975), 240-246

1975

[14] [14]

Math., 13(1) (2024), 35-46

Rahul Jejurkar and Vinayak Joshi,On the comparability graph of lattices, Palestine J. Math., 13(1) (2024), 35-46

2024

[15] [15]

Kelly and I

D. Kelly and I. Rival,Crowns, fences and dismantlable lattices, can. J. Math., 26(5) (1974), 1257-1271

1974

[16] [16]

Maeda and S

F. Maeda and S. Maeda,Theory of symmetric lattices, Springer-Verlag New York Heidelberg Berlin (1970)

1970

[17] [17]

Ore,Theory of graphs, American Mathematical Society Colloquium Publication, 38 (1962), 270 pages

O. Ore,Theory of graphs, American Mathematical Society Colloquium Publication, 38 (1962), 270 pages

1962

[18] [18]

Stern,On the covering graph of balanced lattices, Discrete Math., 156 (1996), 311-316

M. Stern,On the covering graph of balanced lattices, Discrete Math., 156 (1996), 311-316

1996

[19] [19]

Ward,A characterization of Dedekind structures, Bull

M. Ward,A characterization of Dedekind structures, Bull. Amer. Math. Soc., 45 (1939), 448-451

1939

[20] [20]

D. B. West,Introduction to graph theory, Pearson Education, Inc., (2001)

2001