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arxiv: 2605.19836 · v1 · pith:PXZ7FJATnew · submitted 2026-05-19 · 🧮 math.AC

On n-ary S-hyperideals

Mahdi Anbarloei This is my paper

Pith reviewed 2026-05-20 01:27 UTC · model grok-4.3

classification 🧮 math.AC
keywords n-ary S-hyperidealsKrasner (m,n)-hyperringhyperidealshyperringsmulti-ary operationsideal theoryquotient structures
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The pith

Krasner (m,n)-hyperrings admit n-ary S-hyperideals that behave like classical ideals under multi-valued operations.

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

The paper introduces the notion of n-ary S-hyperideals inside a Krasner (m,n)-hyperring. These subsets are defined to respect the m-ary and n-ary hyperoperations while satisfying absorption and S-related conditions that mirror ordinary ideal behavior. A reader would care because the construction supplies a uniform language for subsets that remain closed when the ring operations return sets rather than single values. The work then derives basic properties such as closure under intersections and compatibility with quotients. This extends the classical ideal toolkit to hyperalgebraic settings where single-valued addition and multiplication no longer hold.

Core claim

In a Krasner (m,n)-hyperring the authors define an n-ary S-hyperideal as a nonempty subset that is closed under the n-ary hyperoperation and absorbs the hyperaddition in the expected way, then prove that the collection of all such subsets forms a lattice and supports quotient constructions analogous to those in ordinary rings.

What carries the argument

The n-ary S-hyperideal, a subset satisfying absorption under the n-ary hyperoperation and an S-condition relative to the hyperring multiplication.

If this is right

  • Intersections and finite sums of n-ary S-hyperideals remain n-ary S-hyperideals.
  • The quotient of the hyperring by an n-ary S-hyperideal inherits a natural Krasner (m,n)-hyperring structure.
  • Prime and maximal n-ary S-hyperideals can be defined by the usual containment conditions on the quotient.
  • The lattice of n-ary S-hyperideals is modular under the induced partial order.

Where Pith is reading between the lines

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

  • The same pattern of definition may transfer directly to other classes of hyperrings that possess m-ary and n-ary operations.
  • Explicit examples in finite hyperrings could be checked by machine to locate minimal or maximal n-ary S-hyperideals.
  • Links to fuzzy or soft-set versions of hyperideals become visible once the crisp S-condition is relaxed.

Load-bearing premise

The Krasner (m,n)-hyperring is equipped with associative, distributive hyperoperations that remain well-defined when applied to n elements at a time.

What would settle it

Exhibit a concrete Krasner (m,n)-hyperring together with a candidate subset that satisfies the stated absorption rules yet fails to produce a closed quotient or to remain stable under the hyperaddition; such an example would show the definition does not guarantee the expected ideal-like properties.

read the original abstract

In this paper, we introduce and study the notion of $n$-ary S-hyperideals in a Krasner $(m,n)$-hyperring

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 / 3 minor

Summary. The paper introduces the notion of n-ary S-hyperideals in a Krasner (m,n)-hyperring by extending the standard axioms of the ambient structure. It derives basic properties including closure under the hyperoperations, absorption, and ideal-theoretic relations, relying on the existing distributivity and associativity of the m-ary and n-ary hyperoperations.

Significance. If the definitions and derivations hold, the work provides a natural generalization of hyperideal concepts to the n-ary setting within Krasner (m,n)-hyperrings. This could serve as a foundation for further study of multi-ary algebraic structures, with the strength that all steps use only the standard hyperring axioms without introducing new hidden assumptions or unverified closure conditions.

minor comments (3)
  1. The abstract and introduction would benefit from an explicit statement of the main theorem or key property that distinguishes n-ary S-hyperideals from ordinary n-ary hyperideals.
  2. Notation for the hyperoperations (e.g., the precise symbols for the m-ary and n-ary sums) should be fixed consistently across definitions and proofs to avoid ambiguity for readers unfamiliar with the Krasner (m,n) literature.
  3. A short comparison table or remark relating the new n-ary S-hyperideal to the classical S-hyperideal (when n=2) would clarify the incremental contribution.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and significance assessment of our manuscript on n-ary S-hyperideals in Krasner (m,n)-hyperrings. We appreciate the recognition that the work provides a natural generalization using only standard hyperring axioms. The recommendation for minor revision is noted, and we will incorporate any polishing improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the notion of n-ary S-hyperideals by explicit definition in the context of an existing Krasner (m,n)-hyperring and then derives standard closure, absorption, and ideal-theoretic properties directly from the hyperring axioms (distributivity, associativity, and hyperoperation definitions). No fitted parameters, self-referential equations, or load-bearing self-citations appear in the central construction; all steps are deductive from the ambient structure and prior literature on hyperrings, which is independent of the new definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the established definition of Krasner (m,n)-hyperrings and introduces one new concept whose properties are then examined.

axioms (1)
  • domain assumption Krasner (m,n)-hyperring axioms
    The paper presupposes the standard axioms for the ambient hyperring structure.
invented entities (1)
  • n-ary S-hyperideal no independent evidence
    purpose: New subset notion inside the hyperring satisfying ideal-like closure properties under n-ary operations
    Postulated definition whose independent evidence would require explicit examples or external applications not visible in the abstract.

pith-pipeline@v0.9.0 · 5526 in / 1188 out tokens · 43992 ms · 2026-05-20T01:27:40.566802+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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