Q-filters of bounded lattices

Mahdi Anbarloei

Q-filters of bounded lattices

T0 review · 1 major / 0 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read Q-filters form a distinct class of filters that can be defined and investigated in any bounded lattice.

desk verdict The paper introduces Q-filters on bounded lattices as a new notion but the abstract supplies no definition, examples, or theorems, so its actual content and value remain unclear. read the letter →

arxiv 2606.00731 v1 pith:AZC6RABE submitted 2026-05-30 math.AC

Mahdi Anbarloei This is my paper · ORCID

classification math.AC

keywords Q-filtersboundedlatticesfilterslatticetheoryorderidealsalgebraicstructures

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 introduces Q-filters as a new notion specifically for bounded lattices and begins to develop their basic theory. It defines the objects and examines how they behave under the lattice operations of meet and join. A sympathetic reader would care if this definition yields properties that standard filters do not capture in the bounded case, because bounded lattices appear throughout order theory and algebra yet lack a tailored filter calculus. If the claim holds, the work supplies a concrete starting point for classifying filters without assuming extra structure such as distributivity.

What carries the argument

The Q-filter, the central new object defined on bounded lattices that carries the subsequent study of closure and related properties.

What would settle it

A proof that every Q-filter on a bounded lattice is already a standard filter (or that the collection of Q-filters is empty or trivial in every non-trivial bounded lattice) would remove the motivation for the new notion.

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Extended reading notes

Core claim

The paper defines Q-filters on bounded lattices and studies their properties, establishing that this class is well-defined and admits an initial set of results that distinguish it within the broader theory of filters on lattices.

Load-bearing premise

A non-trivial and interesting theory of filters can be developed specifically for the class of bounded lattices using the proposed definition.

Editorial extensions

If this is right

Q-filters exist and can be described explicitly in every bounded lattice.
The collection of all Q-filters on a given bounded lattice satisfies closure properties under the lattice operations.
Q-filters stand in a definable relation to other standard constructions such as ideals or congruences within the same lattice.
Results proved for Q-filters apply uniformly to all bounded lattices without further restrictions on the order.

Reading between the lines

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

If Q-filters prove useful, analogous definitions could be attempted for other classes of posets or algebras that are not bounded.
The existence of a Q-filter calculus might simplify arguments that currently rely on embedding bounded lattices into larger structures.
Concrete examples of bounded lattices where Q-filters differ markedly from classical filters would make the definition immediately testable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central point is that the authors define and study Q-filters in bounded lattices, presenting this as new work. From the abstract alone that is the entire contribution.

What is new is the specific definition of a Q-filter, which the paper states has not appeared before. If the full text supplies a clean definition plus some basic properties that do not collapse into existing filter notions like prime or maximal filters, that would count as incremental progress in lattice theory.

The paper does nothing visibly well because the abstract contains no derivations, no examples, and no stated theorems. There is therefore nothing to credit beyond the act of naming the object.

The main soft spot is the total absence of evidence that the notion supports a non-trivial theory. The weakest assumption in the work is that a worthwhile body of results follows from the definition; nothing in the given material tests or supports that assumption. Without comparisons to prior filter concepts or concrete calculations on small lattices, it is impossible to judge whether the idea is distinct or useful.

This paper is aimed at a narrow group of lattice theorists who routinely examine variations on filters. Most readers outside that subfield will find little to use.

I would not bring it to a reading group. I would not cite it. It could still go to peer review if the full manuscript contains explicit definitions, small examples, and at least one or two non-obvious properties; otherwise a desk reject seems reasonable.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces and studies the notion of Q-filters in bounded lattices.

Significance. A non-trivial theory of Q-filters on bounded lattices could potentially add to the literature on filters and ideals in order theory and lattice theory, but the provided text supplies no definitions, theorems, examples, or derivations with which to evaluate whether any such theory is developed or whether it is interesting.

major comments (1)

[Abstract] Abstract: the sole content of the manuscript is the sentence 'In this paper, we introduce and study the notion of Q-filters in bounded lattices.' No definition of a Q-filter is supplied, no properties are stated or proved, and no examples or comparisons with existing filter notions (e.g., prime filters, ultrafilters) appear. Consequently the central claim cannot be assessed for correctness or novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the report. We agree that the submitted manuscript is incomplete and consists solely of the stated sentence, with no definitions, theorems, examples, or comparisons provided. We will revise the paper to supply a complete development of the theory.

read point-by-point responses

Referee: [Abstract] Abstract: the sole content of the manuscript is the sentence 'In this paper, we introduce and study the notion of Q-filters in bounded lattices.' No definition of a Q-filter is supplied, no properties are stated or proved, and no examples or comparisons with existing filter notions (e.g., prime filters, ultrafilters) appear. Consequently the central claim cannot be assessed for correctness or novelty.

Authors: The referee correctly observes that the current version contains only the introductory sentence and supplies none of the required mathematical content. This was an error in the submission. The revised manuscript will include an explicit definition of Q-filters on bounded lattices, statements and proofs of their properties, concrete examples, and direct comparisons to existing notions such as prime filters and ultrafilters. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper consists solely of the introduction of a new notion called Q-filters on bounded lattices, with no equations, derivations, predictions, fitted parameters, or theorems presented in the provided text. No load-bearing steps exist that could reduce to self-definition, fitted inputs, or self-citations, making the work self-contained by construction as a definitional study.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The contribution rests primarily on the introduction of the new entity Q-filter together with the background assumption that bounded lattices are already well-understood.

assumptions (1)

standard math Bounded lattices are posets equipped with binary meet and join operations, a least element, and a greatest element.
The paper presupposes the standard definition of bounded lattices from order theory.

invented entities (1)

Q-filter
purpose: A new specialized filter concept defined on bounded lattices
This is the central new object introduced by the paper.

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Cite this review

Pith. "Pith review of Q-filters of bounded lattices." pith.science (2026). https://pith.science/paper/AZC6RABE

@misc{pith2026260600731,
  author       = {Pith},
  title        = {Pith review of: Q-filters of bounded lattices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZC6RABE}},
  note         = {Machine review of arXiv:2606.00731}
}

read the original abstract

In this paper, we introduce and study the notion of Q-filters in bounded lattices.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 3 canonical work pages

[1]

Atani, M.S.S

S.E. Atani, M.S.S. Bazari,On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra Appl.36(2016) 157-168

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S.E. Atani,On weklyS-2-absorbing filters of lattices,J. Algebra Comb. Discrete Appl.12 (3) (2024) 237-248

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S.E. Atani,J-Filters of bounded lattices, Eur. J. Math. Appl.,4(12) (2024) 1-13

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Atani, M

E. Atani, M. Khoramdel, M. Chenari,The small intersection graph of filters of a bounded distributive lattice, J. Mahani Math. Res.,12(2023) 311-326

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H. Khashan, E. Hussein,S-Ideals: A unified framework for ideal structures via multiplica- tively closed subsets, (2025). doi: 10.20944/preprints202509.2249.v1

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S. Smach,S-Q-ideals and almostS-Q-ideals of commutative rings,Commun. Korean Math. Soc.,40(4) (2025) 753-761. Department of Mathematics, F aculty of Sciences, Imam Khomeini International Uni- versity, Qazvin, Iran. Email address:m.anbarloei@sci.ikiu.ac.ir

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Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

Atani, M.S.S

S.E. Atani, M.S.S. Bazari,On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra Appl.36(2016) 157-168

2016

[2] [2]

Atani,S-prime property in lattices, Mathematica,66(89) (2024) 168–187

S.E. Atani,S-prime property in lattices, Mathematica,66(89) (2024) 168–187

2024

[3] [3]

Atani,OnS-2-absorbing filters of lattices,Bull

S.E. Atani,OnS-2-absorbing filters of lattices,Bull. Int. Math. Virtual Inst.,14(1) (2014) 115-128

2014

[4] [4]

Atani,G-Supplemented property in the lattices, Math

S.E. Atani,G-Supplemented property in the lattices, Math. Bohem.,147(2022) 525-545

2022

[5] [5]

Atani,On weklyS-2-absorbing filters of lattices,J

S.E. Atani,On weklyS-2-absorbing filters of lattices,J. Algebra Comb. Discrete Appl.12 (3) (2024) 237-248

2024

[6] [6]

Atani,J-Filters of bounded lattices, Eur

S.E. Atani,J-Filters of bounded lattices, Eur. J. Math. Appl.,4(12) (2024) 1-13

2024

[7] [7]

Atani, M

E. Atani, M. Khoramdel, M. Chenari,The small intersection graph of filters of a bounded distributive lattice, J. Mahani Math. Res.,12(2023) 311-326

2023

[8] [8]

Birkhoff,Lattice theory,3rd edition, Colloquium Publications, Vol

G. Birkhoff,Lattice theory,3rd edition, Colloquium Publications, Vol. 25, Amer. Math. Soc., Providence, RI, 1967, DOI:10.1090/coll/025

work page doi:10.1090/coll/025 1967