Rota-type operators on 2-dimensional dendriform algebras

Bouzid Mosbahi; Imed Basdouri

REVIEW 4 major objections 5 minor 2 cited by

Rota-type operators on 2-dimensional dendriform algebras

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper gives complete matrix lists of Rota-Baxter, Reynolds, Nijenhuis, and averaging operators on all twelve 2-dimensional complex dendriform algebras.

desk verdict The central tables are wrong: Dend6^2 already contradicts the paper's own defining equation, and the weight-λ convention is nonstandard without comment. read the letter →

arxiv 2411.15358 v2 pith:7YFDXKIE submitted 2024-11-22 math.RA

Imed Basdouri , Bouzid Mosbahi This is my paper · ORCID

classification math.RA MSC 17A3017B3816W2016S50

keywords Rota-BaxteroperatorReynoldsNijenhuisaveragingdendriformalgebratwo-dimensionalmatrixclassificationweight

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to give the complete classification of four families of Rota-type operators\u2014Rota-Baxter, Reynolds, Nijenhuis, and averaging\u2014on every 2-dimensional complex dendriform algebra. It starts from the known list of twelve isomorphism classes and writes each linear operator as a $2\times2$ matrix; substituting the matrix into the defining identities turns the operator conditions into polynomial equations in the four entries. The displayed tables in Section 2 are the claimed solutions of these equations, with parameter restrictions recorded. The point of the exercise is to provide an explicit, low-dimensional catalogue that can be used to test and build examples in the wider theory of dendriform algebras.

What carries the argument

The load-bearing object is the system of defining identities: (4)\u2013(5) for a Rota-Baxter operator of weight $\lambda$, (6)\u2013(7) for a Reynolds operator, (8)\u2013(9) for a Nijenhuis operator, and (10)\u2013(11) for an averaging operator. These equations are imposed on a linear endomorphism represented by a $2\times2$ matrix over the basis $\{e_1,e_2\}$. Checking the four basis pairs $(e_1,e_1),(e_1,e_2),(e_2,e_1),(e_2,e_2)$ yields polynomial equations in the matrix entries $a_{11},a_{12},a_{21},a_{22}$; solving these equations is what produces the families in the tables.

What would settle it

Take $Dend_1^2$ with $e_1\prec e_1=e_1$, $e_1\succ e_2=e_2$, and form the operator $P(e_1)=0$, $P(e_2)=-e_2$. Under the Rota-Baxter equations with the standard weight term $\lambda u\succ v$ in the second identity, this $P$ satisfies all four basis-pair equations, so Theorem 2.2's zero-only list is not convention-independent.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the operator search on the twelve algebras can be completed and presented as finite tables. For $Dend_1^2$, Theorem 2.1 gives the weight-0 Rota-Baxter operators as the one-parameter family with $a_{12}$ arbitrary and all other matrix entries zero, Theorem 2.2 gives only the zero matrix for weight 1, and Theorems 2.3\u20132.5 list Reynolds, Nijenhuis, and averaging operators, with the same three diagonal and triangular forms appearing in the Reynolds and averaging cases. The rest of Section 2 extends this pattern to $Dend_2^2(\alpha)$ through $Dend_{12}^2$, recording for each algebra which matrices satisfy the four defining identities. The tables are the result: a complete enumeration of Rota-Baxter operators of weights 0 and 1, and of Reynolds, Nijenhuis, and averaging operators, on each algebra.

Load-bearing premise

The weight-1 classifications rest on the convention written in equation (5), where the extra term is $\lambda u\prec v$ in the $\succ$-identity, and on the correctness of the computations, not displayed in the text, that produce the entries in the Section 2 tables.

Editorial extensions

If this is right

For $Dend_1^2$, the weight-0 Rota-Baxter operators are exactly the one-parameter family with $a_{12}$ arbitrary and all other entries zero, and the weight-1 operators are the zero matrix under the paper's convention.
For each of the twelve algebras, verifying whether a given matrix is a Rota-Baxter, Reynolds, Nijenhuis, or averaging operator is reduced to evaluating the listed identities on four basis pairs.
The same list can be used as input for deformation, cohomology, and representation questions about dendriform algebras, since it provides the complete low-dimensional space of these operators.
The parameter restrictions recorded in the tables indicate where the operator families degenerate, which matters when counting or parameterizing solutions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not explore how the answer changes if the weight term in equation (5) is read as $\lambda u\succ v$, the convention used in the references it cites; redoing the substitution for the weight-1 tables under that convention is the natural next check.
The paper does not discuss 3-dimensional cases, but the same four-pair substitution method would extend there once a classification of 3-dimensional dendriform algebras is available; the present tables would serve as the dimension-2 base case.
The parameter restrictions in the tables can be read as describing low-dimensional solution varieties; computing their irreducible components over $\mathbb{C}$ would give a structural picture of how these operator families sit inside the space of all linear maps on each algebra.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Thanks for the note. I read the paper and agree with your take: the main classification is invalid, and the counterexample is real.

For Dend6^2, products are e1≺e1=e1 and e2≻e2=e2. The weight-0 Rota–Baxter table lists P=[[0,0],[0,a22]] with no restriction. Take u=v=e2 in the ≻-equation (5). The left side is (a22e2)≻(a22e2)=a22^2e2. The right side is P(a22e2≻e2 + e2≻a22e2)=P(2a22e2)=2a22^2e2. So a22^2=2a22^2, forcing a22=0. The listed family has no nonzero operator. This is not a convention dispute; it fails under the paper's own definitions.

There is also a definitional problem. Equation (5) puts λu≺v in the ≻-equation, whereas the standard dendriform Rota–Baxter convention uses λu≻v. The paper never flags the departure, and the weight-1 results depend on it. The claim that "weight 0 implies weight 1" is also false under either convention.

To give credit where it is due: the paper targets a real gap. No one seems to have written down Rota-type operators on these twelve 2-dimensional dendriform algebras, and a correct reference table would be useful bookkeeping. The authors cite the algebra classification [3] and organize by operator type clearly.

But the execution is not close. The one proof that is shown, Theorem 2.1, is internally inconsistent: the case analysis concludes "a12=0 or a21=0" but the theorem then lists only a21=0 with a12 arbitrary. The other eleven algebras are covered by unshown table entries, several of which are wrong. The references contain duplicates and an uncited block [13–35] in the middle of the paper; the acknowledgment to a referee in an arXiv preprint suggests it was pasted from a journal submission.

Bottom line: this is a niche bookkeeping paper whose central product is unreliable. A serious editor should desk reject it in the current form. The only path forward is to recompute every table with the standard definition, show the computations, and verify by substitution. If that were done, the paper could become a minor but usable reference. As it stands, I would not cite it and would not bring it to reading group.

Referee Report

4 major / 5 minor

Summary. The paper claims to give a complete description of Rota-Baxter operators (of weights 0 and 1), Reynolds operators, Nijenhuis operators, and averaging operators on each of the twelve 2-dimensional complex dendriform algebras from the classification of [3]. After setting up a matrix convention, it proves one or two cases explicitly and presents the remaining classifications as tables in Sections 2.1-2.4. The central claim is that these tables list exactly the operators of each type.

Significance. The classification problem is natural and a correct table would be a useful reference for the dendriform-algebra literature. However, direct substitution into the paper's own defining equations contradicts at least two rows of the weight-0 Rota-Baxter table and also the Reynolds list for Dend1^2. No machine-checked proofs or computational artifacts are supplied, so the unverified tables cannot compensate for the demonstrated failures. The dependence on [3] for the algebra list is standard practice and is not a concern; the concern is that the paper's own central classifications are false.

major comments (4)

[Section 2.1, weight-0 Rota-Baxter tables] The listed operators do not satisfy the paper's own equation (5). For Dend6^2, whose products are e1≺e1=e1 and e2≻e2=e2, the table lists P=[[0,0],[0,a22]] with no restriction. Taking u=v=e2 in (5) at weight 0 gives a22^2 e2 = 2a22^2 e2, hence a22=0, so the table's unrestricted family contains no nonzero operator. For Dend4^2, the row P1=[[a11,0],[0,0]] with a11≠0 gives, at u=v=e1, a11^2 e1 = 2a11^2 e1, forcing a11=0. These are direct contradictions of the claimed complete lists and invalidate the central classification claim.
[Section 2.2, Theorem 2.3] The Reynolds-operator list for Dend1^2 contradicts equation (6). For P3=[[a22,0],[0,a22]], substituting u=v=e1 into (6) yields a22^2 e1 = (2a22^2 - a22^3)e1, so a22=0 or a22=1; for P2=[[a11,0],[0,0]] the same substitution forces a11=0 or a11=1. The theorem lists both families without these restrictions, and the proof itself derives a11=0 or 1 before declaring a11 arbitrary. Thus the Reynolds classification is also incorrect as stated.
[Section 1, Eq. (5)] The weight-λ Rota-Baxter definition puts λu≺v in the ≻-equation, whereas the standard dendriform convention used in the cited references [10,11] puts λu≻v there. This is load-bearing: the zero-only weight-1 classification for Dend1^2 in Theorem 2.2 is an artifact of the nonstandard equation. Under the standard convention, the operator P(e1)=0, P(e2)=-e2 satisfies the weight-1 Rota-Baxter equations on Dend1^2. The authors must either adopt the standard definition or explicitly introduce and justify a new one.
[Section 1, after Eq. (5)] The assertion that a weight-0 Rota-Baxter operator is automatically a weight-1 operator is false. For example, the nonzero weight-0 operator P1 in Theorem 2.1, P(e1)=a e2, P(e2)=0, fails the weight-1 equations at u=v=e1 because the right-hand side acquires the nonzero term P(e1)=a e2 while the left-hand side is 0. Consequently the claimed reduction to weights 0 and 1 is not established, and the statement is in tension with the paper's own Theorems 2.1 and 2.2.

minor comments (5)

[Section 2.1, proof of Theorem 2.1] In Case 2, the line '0 = a12 a21' and the following derivation do not follow from the preceding equations; the proof as written does not support the final matrix, even if the final statement happens to be correct.
[Section 2.2, proof of Theorem 2.3] After Case 2 the text states a12=0, yet the theorem lists P1 with arbitrary a12; this internal inconsistency should be repaired.
[Section 2.4, proof of Theorem 2.5] For P3, the displayed calculation of P3(e1)≺P3(e2) uses a22e1, but in Dend1^2 one has e1≺e2=0, and the ≻-calculation omits a factor a22^2. The diagonal operators are valid averaging operators, but the displayed algebra is wrong.
[Introduction, matrix convention] The matrix convention used in the paper, in which the rows of the displayed 2×2 matrix are the coefficients of P(e1) and P(e2), is nonstandard and should be stated explicitly; as written, the table entries are easy to misread.
[References and Conflict of Interests] The reference list contains duplicate entries (for example, [11] and [12] are the same arXiv preprint), a large block of self-citations, and unrelated papers such as the hydrokinetic-turbine articles [26]-[30]. The 'Conflict of Interests' paragraph consists of a citation block [13-35] rather than a conflict statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the classifications are obtained by solving the defining operator equations, not by assuming the target.

full rationale

The derivations in this paper are self-contained with respect to the claims being tested: each family of Rota-Baxter, Reynolds, Nijenhuis, and averaging operators is obtained by substituting a general 2x2 matrix into the defining identities (4)-(11), writing the resulting polynomial equations in the entries, and solving them; the listed matrices are then asserted to satisfy those equations. The target classification does not appear as an input to the solving process, and no fitted parameter is relabeled as a prediction. The only external input is the classification theorem 1.1, quoted from reference [3], which supplies the list of twelve two-dimensional dendriform algebras; using an external classification as a starting point is standard practice and not circular. Reference [3] is not authored by the current authors, and no self-citation is load-bearing: the block of references [13]-[35] appears after the conclusion, is not invoked anywhere in the proofs, and does not enter the derivation chain. The nonstandard weight term lambda u preceq v in equation (5), and the apparent substitution errors in the tables, are mathematical correctness concerns, not circularity: a wrong equation or a wrong computation can invalidate a theorem without making the argument circular. Under the paper's stated definitions, the classification reduces to solving those equations by construction, so there is no circular reduction to report.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters or invented entities are involved. The a_ij entries in the tables are solution variables parameterizing the operator families, not constants fitted to data. The load-bearing assumptions are the completeness of the algebra classification [3] and the correctness of the Rota-Baxter definition in eq. (5).

assumptions (3)

domain assumption Theorem 1.1: the list of twelve 2-dimensional complex dendriform algebras is complete and pairwise non-isomorphic.
Invoked in Section 1 as the foundation for all classifications; taken from [3] without proof.
ad hoc to paper Equation (5) is the correct definition of a weight-λ Rota-Baxter operator with weight term λ u ≺ v in the ≻-equation.
This deviates from the standard definition in the cited references [10,11], which use λ u ≻ v; the deviation is unflagged and changes the weight-1 results.
standard math Reynolds, Nijenhuis, and averaging operator equations (6)-(11) are the intended operator definitions.
These match the standard definitions in the literature; they are assumed without proof.

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Cite this review

Pith. "Pith review of Rota-type operators on 2-dimensional dendriform algebras." pith.science (2026). https://pith.science/paper/7YFDXKIE

@misc{pith2026241115358,
  author       = {Pith},
  title        = {Pith review of: Rota-type operators on 2-dimensional dendriform algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7YFDXKIE}},
  note         = {Machine review of arXiv:2411.15358}
}

abstract

We describe Rota-Baxter operators, Reynolds operators, Nijenhuis operators, and Averaging operators on 2-dimensional dendriform algebras over $\mathbb{C}$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Computational Approaches to Derivations and Automorphism Groups of Associative Algebras
math.RA 2025-01 reject novelty 4.0 of 10

The paper claims to compute derivations and automorphism groups of low-dimensional associative algebras over C, but the computations are not self-contained and contain errors.
An Algorithmic Approach to Inner Derivations of Low-Dimensional Zinbiel Algebras
math.RA 2024-12 reject novelty 2.0 of 10

The central claim that ad_w(u)=u∘w-w∘u is a derivation of every Zinbiel algebra is false; a counterexample appears in the paper's own four-dimensional table.

Reference graph

Works this paper leans on

33 extracted references · 11 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [3]

M., Rakhimov, I

Rikhsiboev, I. M., Rakhimov, I. S., & Basri, W. I. T. R. I. A. N. Y. (2010). The description of dendriform algebra structu res on two-dimensional complex space. Journal of Algebra, Number Theory: Advances and Applicatio ns, 4(1), 1-18. IMED BASDOURI 1, BOUZID MOSBAHI 2 11

work page 2010

[2] [1]

Loday, J. L. (1993). Une version non commutative des alg` e bres de Lie: les alg` ebres de Leibniz. Les Rencontres Physiciens- Math´ ematiciens de Strasbourg-RCP25, 44, 127-151

work page 1993

[3] [2]

L., Chapoton, F., Frabetti, A., Goichot, F

Loday, J. L., Chapoton, F., Frabetti, A., Goichot, F. (200 1). Dialgebras (pp. 7-66). Springer Berlin Heidelberg

work page

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A., & Fiidow, M

Alkhezi, Y. A., & Fiidow, M. A. (2019). On Derivations of Fi nite Dimensional Dendriform Algebras. Pure Mathematical sciences, 8, 17-24

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[5] [5]

Alkhezi, Y. A. (2023). ( ρ, τ, σ )-Derivations of Dendriform Algebras. Applied Mathematic s, 14(12), 839-846

work page 2023

[6] [6]

A., & Fiidow, M

Alkhezi, Y. A., & Fiidow, M. A. (2022). Inner Derivations o f Finite Dimensional Dendriform Algebras. In Internationa l Mathematical Forum (Vol. 17, No. 4, pp. 163-170)

work page 2022

[7] [7]

Zoba, M. A. D. (2018). Centroids Of Dendriform Algebras. I nternational Journal of Pure and Applied Mathematics, 119(15), 427-435

work page 2018

[8] [8]

Zoba, M. A. D. (2021). Quasi-Centroids Of Dendriform Alge bras. PSYCHOLOGY AND EDUCATION, 58(2), 6684-6689

work page 2021