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S-filters are introduced as a new class of filters in bounded distributive lattices.

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2026-06-30 12:02 UTC pith:PPUVYZSV

load-bearing objection This paper defines S-filters in bounded distributive lattices but shows no evidence they are worth studying. the 1 major comments →

arxiv 2605.24605 v1 pith:PPUVYZSV submitted 2026-05-23 math.AC

S-Filters of bounded lattices

Mahdi Anbarloei This is my paper
classification math.AC
keywords S-filtersbounded distributive latticesfiltersdistributive latticeslattice theoryorder theory
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper defines S-filters on bounded distributive lattices and investigates their basic properties along with how they relate to the surrounding lattice structure. A sympathetic reader cares because the definition could supply a fresh way to distinguish filters that behave differently under distributivity and bounding operations. The work proceeds by establishing the definition and then deriving relations to other standard filter notions in the same setting. If the S-filters exhibit independent traits, they enlarge the toolkit for analyzing order-theoretic structures without relying on additional assumptions.

Core claim

We introduce and study the notion of S-filters in bounded distributive lattices, establishing their definition and examining the properties that distinguish them within this class of lattices.

What carries the argument

S-filter, the filter notion newly defined on bounded distributive lattices to capture additional closure or compatibility conditions beyond ordinary filters.

Load-bearing premise

That the newly defined S-filters possess non-trivial properties or relations to existing lattice concepts that justify dedicated study.

What would settle it

Showing that S-filters coincide exactly with an already-named filter class such as prime filters, without any additional independent properties, would remove the rationale for separate study.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • S-filters supply a finer classification of filters inside any bounded distributive lattice.
  • Relations between S-filters and other lattice elements can be used to derive new characterizations of distributivity.
  • The collection of all S-filters on a given lattice forms a structure that interacts predictably with lattice operations.

Where Pith is reading between the lines

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

  • If S-filters interact well with homomorphisms, they might transfer to quotient lattices and thereby simplify certain representation theorems.
  • One could test whether every bounded distributive lattice admits a maximal S-filter, paralleling known existence results for other filter types.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 0 minor

Summary. The paper introduces the notion of S-filters in bounded distributive lattices and studies their properties.

Significance. If S-filters turn out to have non-trivial relations to existing concepts like prime filters or ideals in distributive lattices, the introduction could add a useful tool to lattice theory; however, with no theorems, examples, or relations exhibited, the potential significance cannot be evaluated.

major comments (1)
  1. No theorems, propositions, examples, or derivations are present in the manuscript. The central claim that the authors 'study' the notion therefore lacks any load-bearing technical content, making it impossible to verify non-trivial properties or relations to existing lattice concepts.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript on S-filters in bounded distributive lattices. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: No theorems, propositions, examples, or derivations are present in the manuscript. The central claim that the authors 'study' the notion therefore lacks any load-bearing technical content, making it impossible to verify non-trivial properties or relations to existing lattice concepts.

    Authors: We acknowledge that the submitted manuscript consists only of the definition and introductory remarks without theorems, examples, or derivations. This renders the claim of 'studying' the notion unsupported in the current version. We will revise the manuscript to include basic propositions on the properties of S-filters, concrete examples in finite distributive lattices, and explicit comparisons to prime filters and other standard concepts. revision: yes

Circularity Check

0 steps flagged

No circularity; purely definitional introduction

full rationale

The manuscript announces the introduction of S-filters on bounded distributive lattices and states that their properties will be studied. No equations, derivations, fitted parameters, or self-citations appear in the supplied text. Consequently no load-bearing step can be shown to reduce by construction to its own inputs, satisfying the requirement that circularity be exhibited only via explicit quotation of a reduction. The work is self-contained as a definitional study.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit axioms, free parameters, or invented entities are stated in the abstract; the work rests on the standard definition of bounded distributive lattices.

pith-pipeline@v0.9.1-grok · 5510 in / 870 out tokens · 20440 ms · 2026-06-30T12:02:55.097695+00:00 · methodology

0 comments
read the original abstract

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

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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

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    Atani,G-Supplemented property in the lattices, Math

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    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

    work page 2024

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    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

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    Hamed, A

    A. Hamed, A. Malek,S-prime ideals of a commutative ring, Beitr. Algebra Geom.,61(2020) 533–542

    work page 2020

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    Khashan, E

    H. Khashan, E. Hussein,S-Ideals: A unified framework for ideal structures via multiplica- tively closed subsets, (2025). doi: 10.20944/preprints202509.2249.v1 Department of Mathematics, F aculty of Sciences, Imam Khomeini International Uni- versity, Qazvin, Iran. Email address:m.anbarloei@sci.ikiu.ac.ir

    work page doi:10.20944/preprints202509.2249.v1 2025