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arxiv: 2605.01007 · v1 · submitted 2026-05-01 · 🧮 math.RA · math.AT

Recognition: unknown

Nonsymmetric versions of binary quadratic operads

B. K. Sartayev, F. A. Mashurov

Pith reviewed 2026-05-09 14:31 UTC · model grok-4.3

classification 🧮 math.RA math.AT
keywords nonsymmetric operadsbinary quadratic operadswhite Manin productassociative operadoperad theorycombinatorial structure
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The pith

A binary quadratic operad Var admits a nonsymmetric version when the white Manin product As∘Var satisfies an explicit structural criterion.

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

The paper defines what it means for a binary quadratic operad to possess a nonsymmetric version. It then derives a criterion that tells exactly when the white Manin product of the associative operad with Var inherits this version. The criterion is checked on concrete examples and counterexamples, after which combinatorial properties are recorded for those operads that pass the test. A reader cares because the construction supplies a systematic way to detect and exploit reduced symmetry in families of operads built from As.

Core claim

The authors introduce the notion of a nonsymmetric version of a binary quadratic operad Var and prove that the operad As∘Var possesses this version precisely when Var satisfies a stated algebraic criterion expressed in terms of the white Manin product. They verify the criterion on several families of operads, exhibit counterexamples where it fails, and, for the cases that succeed, give explicit descriptions of the resulting combinatorial structure.

What carries the argument

The nonsymmetric version of Var, together with the white Manin product As∘Var that carries the criterion.

If this is right

  • Operads that meet the criterion possess explicitly describable combinatorial properties.
  • The same construction supplies both positive examples and counterexamples among standard binary quadratic operads.
  • Symmetry properties of As∘Var are completely determined by the criterion on Var.

Where Pith is reading between the lines

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

  • The criterion may be lifted to other binary operads or to different Manin products once the nonsymmetric notion is fixed.
  • Combinatorial descriptions obtained for successful cases could be used to enumerate basis elements or to compute generating functions for the corresponding operads.
  • The distinction between symmetric and nonsymmetric versions may clarify when an operad admits a set-theoretic or combinatorial model.

Load-bearing premise

The newly defined notion of a nonsymmetric version must be a stable and well-defined property of binary quadratic operads under the white Manin product.

What would settle it

A binary quadratic operad Var for which the stated criterion predicts that As∘Var is nonsymmetric, yet direct computation of the product shows it is not (or the converse).

read the original abstract

In this paper, we study the white Manin product of the associative operad $\As$ with a binary quadratic operad $\Var$. We introduce the notion of a nonsymmetric version of $\Var$ and provide a criterion for determining when the operad $\As\circ\Var$ has this property. We illustrate the construction with several examples and counterexamples. Finally, for some operads admitting nonsymmetric versions, we describe their combinatorial properties.

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 / 2 minor

Summary. The paper studies the white Manin product of the associative operad As with a binary quadratic operad Var. It introduces the notion of a nonsymmetric version of Var and provides a criterion for determining when the operad As∘Var has this property. The construction is illustrated with several examples and counterexamples, and for some operads admitting nonsymmetric versions the paper describes their combinatorial properties.

Significance. If the criterion holds, the work supplies a concrete tool for analyzing nonsymmetric structures arising from the white Manin product, extending existing operad theory in a useful direction. Credit is due for the sequence of concrete examples and counterexamples that directly test the criterion, together with the combinatorial descriptions supplied for selected cases; these elements make the results falsifiable and applicable in algebraic combinatorics.

minor comments (2)
  1. [Abstract] The abstract states that the criterion is illustrated with 'several examples and counterexamples' but does not name the specific operads Var that are treated; adding one or two explicit instances (e.g., the operad of Lie or Jordan algebras) would improve immediate readability.
  2. Notation for the white Manin product alternates between As∘Var (plain text) and As∘Var (LaTeX); a uniform choice of symbol throughout the manuscript would eliminate minor ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on nonsymmetric versions of binary quadratic operads via the white Manin product As ∘ Var, the criterion provided, and the value placed on the examples, counterexamples, and combinatorial descriptions. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a new definition of a 'nonsymmetric version' for binary quadratic operads and states an explicit criterion for when the white Manin product As∘Var satisfies the property. This is foundational definitional work accompanied by concrete examples and counterexamples that test the criterion directly. No derivations reduce by construction to fitted inputs, self-citations, or renamed known results; the central claim rests on the new definition itself rather than any circular reduction. The argument is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the newly introduced definition of nonsymmetric version and the criterion for the Manin product; these are not derived from prior results but postulated as new objects.

axioms (1)
  • domain assumption Standard axioms and definitions of binary quadratic operads and the white Manin product from prior operad theory.
    The paper assumes familiarity with and correctness of existing operad constructions.
invented entities (1)
  • nonsymmetric version of a binary quadratic operad no independent evidence
    purpose: To capture a symmetry-free property of operads under Manin products.
    Newly defined concept whose independent existence or utility outside this paper is not established in the abstract.

pith-pipeline@v0.9.0 · 5361 in / 1130 out tokens · 34655 ms · 2026-05-09T14:31:00.028083+00:00 · methodology

discussion (0)

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Reference graph

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