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arxiv: 2501.07302 · v2 · submitted 2025-01-13 · 🧮 math.RA

Anti-associative dendriform algebras

Zafar Normatov This is my paper

Pith reviewed 2026-05-23 05:47 UTC · model grok-4.3

classification 🧮 math.RA
keywords anti-associative algebrasdendriform algebrasConnes cocyclesO-operatorsoperadic splittingalgebraic varietiesbinary operations
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The pith

Anti-associative algebras with nondegenerate Connes cocycles admit compatible anti-associative dendriform structures.

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

The paper applies the general method of splitting an algebraic product into two operations to the anti-associative case. It defines anti-associative dendriform algebras as pairs of operations whose sum satisfies the anti-associativity identity. O-operators on anti-associative algebras serve as a tool for constructing these splits. The central result states that any anti-associative algebra carrying a nondegenerate Connes cocycle can be equipped with a compatible anti-associative dendriform structure. A reader would care because the result supplies a concrete decomposition that turns a single anti-associative product into two interacting operations.

Core claim

Anti-associative algebras with nondegenerate Connes cocycles admit compatible anti-associative dendriform algebra structures. These structures arise from two binary operations whose sum is anti-associative, and the O-operator formalism on anti-associative algebras provides the interpretive link between the cocycle and the split operations.

What carries the argument

O-operators on anti-associative algebras, which interpret the splitting of the anti-associative product into a pair of dendriform operations when a nondegenerate Connes cocycle is present.

If this is right

  • Anti-associative algebras carrying nondegenerate Connes cocycles become objects that can be studied through their two-operation dendriform decompositions.
  • The O-operator construction supplies a uniform way to produce new anti-associative dendriform algebras from existing anti-associative algebras.
  • The same operadic splitting technique applies to the anti-associative variety as it does to previously treated varieties.
  • Compatible dendriform structures on these algebras yield new representations and possible cohomology theories derived from the split operations.

Where Pith is reading between the lines

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

  • The result may allow properties such as deformation theory or representations to transfer from the dendriform side back to the original anti-associative algebra.
  • Similar splitting constructions could be tested on other nonassociative identities once the operadic framework is verified for each case.
  • If the Connes cocycle condition can be relaxed or replaced by weaker data, the class of algebras admitting dendriform splits would enlarge.

Load-bearing premise

The operadic splitting framework developed for other algebraic varieties extends directly to the anti-associative identity without additional obstructions.

What would settle it

An explicit example of an anti-associative algebra that possesses a nondegenerate Connes cocycle yet admits no pair of operations summing to the given product and satisfying the dendriform axioms would disprove the central claim.

read the original abstract

The general operadic approach to splitting algebraic operations was developed in \cite{BBGN}. By splitting the product in a given algebraic variety $\mathcal{C}$, notion of $\mathcal{C}$-dendriform algebras was systematically studied in \cite{OPV}. This article aims to study ``anti-associative dendriform algebras", which offer an approach to addressing anti-associativity. These algebras are defined by two operations whose sum is anti-associative. Furthermore, the notion of $\mathcal{O}$-operators on anti-associative algebras is presented as a tool to interpret anti-associative dendriform algebras. Moreover, anti-associative algebras with nondegenerate Connes cocycles admit compatible anti-associative dendriform algebra structures.

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

1 major / 0 minor

Summary. The paper extends the operadic splitting framework of BBGN and OPV to the anti-associative variety. It defines anti-associative dendriform algebras via two operations whose sum satisfies the anti-associative identity (xy)z + x(yz) = 0, introduces O-operators on anti-associative algebras, and claims that any anti-associative algebra equipped with a nondegenerate Connes cocycle admits a compatible anti-associative dendriform structure.

Significance. If the central existence claim holds, the work supplies the first systematic dendriform-type splitting for anti-associative algebras and links it to Connes cohomology, thereby enlarging the class of varieties to which the BBGN/OPV construction applies.

major comments (1)
  1. [Abstract] Abstract (and the statement of the main existence result): the claim that nondegenerate Connes cocycles induce compatible anti-associative dendriform structures rests on direct transfer of the operadic splitting. The anti-associative identity produces sign flips relative to the associative case in the four resulting dendriform-type relations. The manuscript provides no explicit verification that the cocycle-induced O-operator satisfies these signed identities, nor that the bijectivity of the O-operator correspondence survives the sign change. This verification is load-bearing for the central claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires explicit verification to support the central existence claim. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the statement of the main existence result): the claim that nondegenerate Connes cocycles induce compatible anti-associative dendriform structures rests on direct transfer of the operadic splitting. The anti-associative identity produces sign flips relative to the associative case in the four resulting dendriform-type relations. The manuscript provides no explicit verification that the cocycle-induced O-operator satisfies these signed identities, nor that the bijectivity of the O-operator correspondence survives the sign change. This verification is load-bearing for the central claim.

    Authors: We agree that the anti-associative identity introduces sign changes relative to the associative case, and that an explicit check is needed to confirm the cocycle-induced O-operator satisfies the four signed dendriform relations and that bijectivity is preserved. While the general operadic splitting of BBGN/OPV applies formally, the manuscript does not contain this verification. In the revised version we will add a dedicated subsection that (i) derives the signed relations from the anti-associative identity, (ii) substitutes the O-operator defined by the nondegenerate Connes cocycle, and (iii) verifies that the resulting identities hold and that the correspondence remains bijective. This will make the central claim self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; central existence claim rests on external operadic splitting framework from independent citations.

full rationale

The paper defines anti-associative dendriform algebras directly via the sum of two operations being anti-associative and introduces O-operators as an interpretive tool. The key existence statement (anti-associative algebras with nondegenerate Connes cocycles admit compatible structures) is obtained by applying the splitting construction from the cited works BBGN and OPV. These citations are to prior literature by different authors; no self-citation chain is load-bearing, no parameter is fitted and renamed as a prediction, and no equation reduces the new structures to the inputs by definition. The derivation chain therefore remains self-contained against the external benchmarks it invokes.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces one new algebraic structure defined by identities; it relies on the standard axioms of binary algebras and on the operadic framework of the cited works. No numerical parameters or new postulated objects with independent evidence appear.

axioms (1)
  • domain assumption The operadic splitting construction of BBGN and OPV applies verbatim to the anti-associativity identity.
    Invoked in the opening paragraph to justify the definition of the new variety.
invented entities (1)
  • anti-associative dendriform algebra no independent evidence
    purpose: To provide a splitting of an anti-associative product into two operations.
    Newly defined in the abstract; no independent existence proof or external realization is given.

pith-pipeline@v0.9.0 · 5637 in / 1298 out tokens · 27957 ms · 2026-05-23T05:47:52.539553+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Aguiar, Pre-Poisson algebras, Lett

    M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54 (2 000), 263-277

    work page

  2. [2]

    Abdurasulov, J

    K. Abdurasulov, J. Adashev, Z. Normatov, Sh. Solijonova , Classification of three dimensional anti-dendriform alge bras. Comm. Alg. (2024), 1–17

    work page 2024

  3. [3]

    Bai, Double constructions of Frobenius algebras, Con nes cocycles and their duality, J

    C. Bai, Double constructions of Frobenius algebras, Con nes cocycles and their duality, J. Noncommut. Geom. 4 (2010) , 475-530

    work page 2010

  4. [4]

    Baklouti, S

    A. Baklouti, S. Benayadi, A. Makhlouf, S. Mansour, Jacob i–Jordan-admissible algebras and pre-Jacobi–Jordan alge bras, J. Algebra Its Appl. (2025), 2550142

    work page 2025

  5. [5]

    C. Bai, L. Guo and X. Ni, O-operators on associative algeb ras and associative Y ang-Baxter equations, Pacific J. Math.256 (2012), 257-289

    work page 2012

  6. [6]

    Burde, A

    D. Burde, A. Fialowski, Jacobi–Jordan algebras. Linear Algebra Its Appl. 459 (2014), 586–594. [C] F. Chapoton, Un th´ eor` eme de Cartier-Milnor-Moore-Qu illen pour les big` ebres dendriformes et les alg` ebres brac es, J. Pure and Appl. Algebra 168 (2002), 1-18

    work page 2014

  7. [7]

    C. Bai, D. Meng, The classification of left-symmetric alg ebras in dimension 2, Chin. Sci. Bull. 41(23) (1996), 2207

    work page 1996

  8. [8]

    Basri and I.M

    W. Basri and I.M. Rikhsiboev, On low dimensional diassoc iative algebras, Proceedings of Third Conference on Resear ch and Education in Mathematics (ICREM-3), UPM, Malaysia, (2007), 164-170

    work page 2007

  9. [9]

    Ebrahimi-Fard, D

    K. Ebrahimi-Fard, D. Manchon, F. Patras, New identities in dendriform algebras, J. Algebra 320 (2008), 708-727

    work page 2008

  10. [10]

    Fehlberg J´unior, I

    R. Fehlberg J´unior, I. Kaygorodov, C. Kuster, The alge braic and geometric classification of antiassociative alge bras, RACSAM, 116(2) (2022), 78

    work page 2022

  11. [11]

    Haliya, G.D

    C.E. Haliya, G.D. Houndedji, On a pre-Jacobi-Jordan al gebra: relevant properties and double construction, http://arxiv.org/abs/2007.06736v1

    work page arXiv 2007

  12. [12]

    D. Gao, G. Liu, C. Bai, Anti-dendriform algebras, new sp litting of operations and Novikov type algebras, JACO, 59 (2 024), 661-696

    work page

  13. [13]

    Kaygorodov, Non-associative algebraic structures : classification and structure, Commun

    I. Kaygorodov, Non-associative algebraic structures : classification and structure, Commun. Math. 32(3) (2024), 162

    work page 2024

  14. [14]

    Kobayashi, K

    Y . Kobayashi, K. Shirayanagi, S. Takahasi, M. Tsukada, A complete classification of three-dimensional algebras over R and C, Asian-Eur. J. Math. 14(8) (2021), 2150131

    work page 2021

  15. [15]

    Loday, Dialgebras, in Dialgebras and related ope rads, Lecture Notes in Math

    J.-L. Loday, Dialgebras, in Dialgebras and related ope rads, Lecture Notes in Math. 1763 (2002), 7-66

    work page 2002

  16. [16]

    Petersson, The classification of two-dimensional no nassociative algebras, Results Math

    H. Petersson, The classification of two-dimensional no nassociative algebras, Results Math. 37(1-2) (2000), 120- 154

    work page 2000

  17. [17]

    Remm, Anti-associative algebras, http: //arxiv.org/abs/2202.10812v2

    E. Remm, Anti-associative algebras, http: //arxiv.org/abs/2202.10812v2

    work page arXiv

  18. [18]

    I. M. Rikhsiboev, I.S. Rakhimov, W. Basri, Classificati on of 3-dimensional complex diassociative algebras, MJMS. 4(2) (2010), 241-254

    work page 2010

  19. [19]

    Rikhsiboev, I

    I. Rikhsiboev, I. Rakhimov, W. Basri, The description o f dendriform algebra structures on two-dimensional comple x space, JPaNTA. 4(1) (2010), 1-18

    work page 2010

  20. [20]

    Ronco, Primitive elements in a free dendriform algeb ra, In: New Trends in Hopf Algebra Theory (La Falda, 1999), Co ntemp

    M. Ronco, Primitive elements in a free dendriform algeb ra, In: New Trends in Hopf Algebra Theory (La Falda, 1999), Co ntemp. Math. 267 (2000), 245-263

    work page 1999

  21. [21]

    Uchino, Quantum analogy of Poisson geometry, relate d dendriform algebras and Rota-Baxter operators, Lett

    K. Uchino, Quantum analogy of Poisson geometry, relate d dendriform algebras and Rota-Baxter operators, Lett. Mat h. Phys. 85 (2008), 91-109. School of Mathematics, J ilin University, C hangchun, 130012, C hina,, I nstitute of Mathematics, U zbekistan Academy of Sciences, Uni- versity Street, 9, Olmazor district, Tashkent, 100174, Uzbekistan Email address...

    work page 2008