arxiv: 2605.00712 · v1 · submitted 2026-05-01 · 🧮 math.RA

Recognition: unknown

On local function, an algebraic approach

Monoj Kumar Das, Shyamapada Modak

Pith reviewed 2026-05-09 14:41 UTC · model grok-4.3

classification 🧮 math.RA

keywords limit pointspower set ringidealsanti-idealsgroup topologyhomomorphic imagestopological structure

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The pith

Ideals in the power set ring of a group define limit points that combine with anti-ideals to induce a topology, while homomorphic images alter the structure between groups.

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

The paper seeks to define the limit point of a subset in a group by reference to an ideal in the power set ring. When this definition is combined with the notion of an anti-ideal, the pair satisfies the axioms needed to place a topology directly on the group. The construction further shows that the images of the ideal and anti-ideal under a group homomorphism produce a corresponding but transformed topology on the image group. A reader might care because the method supplies an algebraic route to topological concepts such as closure and convergence inside groups.

Core claim

The limit point concept of a subset in a group is obtained via the ideal of the power set ring. This idea along with the anti-ideal gives the topological structure in a group. Homomorphic images of both ideal and anti-ideal play the role of changing the topological structure from one system to another system.

What carries the argument

The ideal of the power set ring that supplies limit points, used together with the anti-ideal to meet the topology axioms on the group.

If this is right

The limit points defined from the ideal satisfy the properties required to generate a topology on the group.
The homomorphic image of the ideal and anti-ideal yields a valid but altered topology on the image group.
This supplies an algebraic mechanism that relates topological features across groups connected by homomorphisms.

Where Pith is reading between the lines

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

The construction could be checked on familiar groups such as the integers under addition to see whether the resulting topology matches any standard one.
The method might be applied to other algebraic objects that possess a power set ring, such as rings or modules, to induce analogous topologies.
It raises the question of which group invariants remain unchanged when the topology is transported by a homomorphism.

Load-bearing premise

Ideals of the power set ring can be used to define limit points whose combination with anti-ideals meets every axiom required for a topology on the group.

What would settle it

A concrete group, subset, and ideal for which the derived limit points fail one of the topology axioms, such as the intersection of two open sets not being open.

read the original abstract

The paper discuss the limit point concept of a subset in a group via ideal of the power set ring. This idea along with anti-ideal give the topological structure in a group. Homomorphic images of both ideal and anti-ideal are played the remarkable role to change the topological structure from one system to another system.

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 sketches a topology on groups via ideals in the power-set ring plus a new 'anti-ideal' notion, with homomorphisms changing the structure, but the claims rest on unshown definitions and unverified axioms.

read the letter

The main thing here is an algebraic construction that defines limit points of subsets in a group using ideals from the Boolean ring on the power set, then uses something called anti-ideals to produce a topology, while homomorphic images are said to produce different topologies on the image group. That combination and the focus on how homomorphisms act on these objects is the part that is not immediately identical to standard filter or ideal-based topologies in the literature I know. The paper does try to give a ring-theoretic handle on topological structures inside groups, which is a direction worth exploring if the details check out. It earns credit for attempting to link the two areas without heavy machinery. The soft spots are more serious. The abstract supplies no definitions for anti-ideals, no derivation showing the resulting collection satisfies the three topology axioms, and no worked example on even a small group such as Z or Z/4Z. Without those, it is impossible to tell whether the construction actually yields a topology or merely a collection that happens to contain the empty set and the whole group. The term anti-ideal looks invented for the paper and is not compared to existing notions like filters or dual ideals, so overlap or redundancy cannot be ruled out. The title mentions local functions, yet the abstract never explains what they are or how they relate to the limit-point construction. The citation pattern is invisible in what was supplied, but the absence of any comparison to known algebraic topologies is a clear gap. This is for a narrow audience of algebraists already working at the ring-topology boundary who might want to tinker with the idea. A reader who needs reproducible examples, checked axioms, or clear novelty relative to prior work will get little value. It does not deserve a serious referee in its present state because the central claims lack the supporting derivations and checks that would let a referee evaluate them.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes defining limit points of subsets in a group via ideals in the Boolean ring of the power set (with symmetric difference and intersection). It introduces the auxiliary notion of an 'anti-ideal' and claims that the resulting family of sets forms a topology on the group. The paper further asserts that homomorphic images of both ideals and anti-ideals induce a changed but still valid topological structure on the target group.

Significance. If the construction were shown to satisfy the topology axioms and the homomorphism claim were proved, the work would supply an algebraic mechanism for generating and transporting topologies on groups using only ring-theoretic data. This could be of interest at the interface of Boolean rings, ideal theory, and topological algebra, especially if concrete examples or comparisons with existing constructions (e.g., profinite topologies or Zariski-type topologies) were supplied.

major comments (3)

[Main construction (after definition of anti-ideal)] The central claim that the ideal-plus-anti-ideal construction yields a topology is asserted in the abstract and presumably developed in the main body, yet no verification is supplied that the resulting collection of sets is closed under arbitrary unions and finite intersections or contains the empty set and the whole group. Without this check the topology assertion remains unproven.
[Definition of anti-ideal] The notion 'anti-ideal' is introduced as a new entity with no prior reference or standard definition. The manuscript must supply an explicit set-theoretic or algebraic definition and prove that it interacts with the ideal-based limit-point operator in a way that produces a topology; absent this, the construction rests on an ad-hoc object whose properties are not independently grounded.
[Homomorphic images section] The claim that homomorphic images of ideals and anti-ideals 'change the topological structure' is stated but not accompanied by a precise statement of what the induced topology on the image group is, nor by a proof that the image family again satisfies the topology axioms. A counter-example or a general theorem is required.

minor comments (2)

[Abstract] The abstract contains grammatical errors ('The paper discuss', 'are played the remarkable role') that should be corrected for readability.
[Preliminaries] Notation for the power-set ring operations (symmetric difference, intersection) and for the limit-point operator should be introduced once and used consistently; currently the text appears to rely on informal description.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and for identifying these gaps in the presentation of our results. We will undertake a major revision to supply the missing explicit definitions, verifications, and proofs.

read point-by-point responses

Referee: [Main construction (after definition of anti-ideal)] The central claim that the ideal-plus-anti-ideal construction yields a topology is asserted in the abstract and presumably developed in the main body, yet no verification is supplied that the resulting collection of sets is closed under arbitrary unions and finite intersections or contains the empty set and the whole group. Without this check the topology assertion remains unproven.

Authors: We agree that the manuscript does not contain an explicit verification of the topology axioms. In the revised version we will insert a proposition that directly checks closure under arbitrary unions, finite intersections, and the presence of the empty set and the full group, using the definitions of limit points via ideals and the auxiliary anti-ideal condition. revision: yes
Referee: [Definition of anti-ideal] The notion 'anti-ideal' is introduced as a new entity with no prior reference or standard definition. The manuscript must supply an explicit set-theoretic or algebraic definition and prove that it interacts with the ideal-based limit-point operator in a way that produces a topology; absent this, the construction rests on an ad-hoc object whose properties are not independently grounded.

Authors: The anti-ideal is introduced in the paper as a new algebraic object complementary to ideals in the Boolean ring of the power set. We will expand its definition in Section 2 to a fully explicit set-theoretic formulation and add a lemma that proves the combined ideal-plus-anti-ideal family satisfies the topology axioms. revision: yes
Referee: [Homomorphic images section] The claim that homomorphic images of ideals and anti-ideals 'change the topological structure' is stated but not accompanied by a precise statement of what the induced topology on the image group is, nor by a proof that the image family again satisfies the topology axioms. A counter-example or a general theorem is required.

Authors: We accept that the homomorphic-image section lacks a precise statement and a complete proof. The revision will define the induced topology on the image group explicitly and prove that the image family satisfies the topology axioms; if the claim requires additional hypotheses we will state them clearly. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs limit points of subsets in a group directly from ideals in the Boolean ring of the power set (symmetric difference and intersection) and combines them with anti-ideals to satisfy the topology axioms on the group, then tracks how homomorphic images of ideals and anti-ideals induce new topologies. This is a self-contained algebraic definition with no fitted parameters, no self-referential equations that reduce the output to the input by construction, and no load-bearing self-citations or uniqueness theorems imported from prior work by the same authors. The derivation chain is independent and consistent with standard constructions of topologies via ring ideals; no step collapses to a renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the standard ring structure of the power set and the new notion of anti-ideal; no numerical parameters are introduced.

axioms (1)

standard math The power set of a group can be equipped with ring operations making it a ring whose ideals are well-defined.
Invoked implicitly when the abstract refers to ideals of the power set ring.

invented entities (1)

anti-ideal no independent evidence
purpose: To complement ideals so that their joint use produces a topology on the group.
The term appears without prior definition or external reference in the abstract.

pith-pipeline@v0.9.0 · 5330 in / 1372 out tokens · 64840 ms · 2026-05-09T14:41:53.439269+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 7 canonical work pages

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