arxiv: 2605.09512 · v1 · submitted 2026-05-10 · 🧮 math.RA

Recognition: 2 theorem links

· Lean Theorem

Varieties of bicommutative algebras with identity of degree three

Bekzat Zhakhayev, Vesselin Drensky

Pith reviewed 2026-05-12 02:38 UTC · model grok-4.3

classification 🧮 math.RA

keywords bicommutative algebraspolynomial identitiesvarieties of algebrasdegree three identitiesdistributive latticesubvarietiescharacteristic zero

0 comments

The pith

Varieties of bicommutative algebras satisfying a degree-three identity are completely described over characteristic-zero fields.

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

The paper gives a full list of all varieties of bicommutative algebras that obey one extra polynomial identity of total degree three. Bicommutative algebras are nonassociative algebras whose multiplication satisfies two specific identities that enforce a limited form of commutativity. Knowing every possible variety at this low degree organizes the possible algebraic structures that can arise under these rules. The work also supplies an exact criterion that tells when the lattice of subvarieties inside one of these varieties is distributive. This matters because it turns the collection of all such algebras into a structured family that can be examined case by case.

Core claim

The authors provide a complete description of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three. They also establish a sufficient and necessary condition for a variety of bicommutative algebras to have a distributive lattice of subvarieties.

What carries the argument

The bicommutative identities together with any additional identity of total degree three, which together determine the possible varieties.

If this is right

Every bicommutative variety with a cubic identity belongs to one of a finite number of explicitly described classes.
The subvariety lattice of any such variety is distributive precisely when the given algebraic condition holds.
The possible cubic identities compatible with the bicommutative laws can be reduced to a short list of standard forms.
Any algebra in one of these varieties satisfies all the higher identities that follow from its defining cubic relation.

Where Pith is reading between the lines

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

The classification supplies a concrete starting point for checking whether a given low-dimensional algebra satisfies a cubic identity.
It may allow systematic comparison of bicommutative varieties with other nonassociative varieties that obey similar low-degree relations.
The distributivity criterion could be tested directly on small examples to see which varieties admit simple lattice structures.

Load-bearing premise

The base field has characteristic zero.

What would settle it

An explicit bicommutative algebra over a field of characteristic zero that satisfies some degree-three identity but is not contained in any of the varieties listed in the classification.

read the original abstract

The variety of bicommutative algebras is the class of all nonassociative algebras satisfying the polynomial identities $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. In this paper we provide a complete description of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three. Furthermore, we establish a sufficient and necessary condition for a variety of bicommutative algebras to have a distributive lattice of subvarieties.

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

This paper classifies bicommutative varieties over char-0 fields that obey one extra degree-3 identity and gives a lattice criterion for when the subvariety lattice is distributive.

read the letter

This paper classifies all varieties of bicommutative algebras over a field of characteristic zero that satisfy one extra polynomial identity of degree three. It also gives a necessary and sufficient condition for the lattice of subvarieties to be distributive. The authors linearize the cubic identities and enumerate the multilinear possibilities modulo the two bicommutative relations, then extract the explicit varieties and the structural test for the lattice. The approach is direct and fits the standard toolkit for low-degree PI-classifications in nonassociative algebras. The bicommutative laws are simple enough that the degree-three case stays finite and manageable, so the enumeration produces a clean list rather than an open-ended problem. The lattice criterion is a useful byproduct that turns the classification into something with a testable property instead of just a catalog. The main limitation is that the argument is a long case-by-case check. Readers will need to follow the enumeration carefully to confirm nothing was missed, though the stress-test finds no gaps or non-exhaustive splits. The work stays in characteristic zero because linearization requires it, so the results do not carry over to positive characteristic without new tools. No machine-checked proofs or independent verification data are supplied. This is a paper for specialists already working on varieties of nonassociative algebras or on subvariety lattices in PI-theory. A reader who knows the bicommutative setting will get immediate use from the explicit forms and the distributivity test. It deserves a serious referee because the claims are concrete, the methods are appropriate, and the results fill a specific gap in an established classification program.

Referee Report

1 major / 0 minor

Summary. The paper claims to provide a complete description of all varieties of bicommutative algebras over a field of characteristic zero that satisfy an additional polynomial identity of total degree three. It further establishes a necessary and sufficient condition for such a variety to possess a distributive lattice of subvarieties.

Significance. If the classification and criterion are correct, the work would constitute a useful contribution to the PI-theory of nonassociative algebras by exhausting the degree-3 case for bicommutative algebras via linearization and representation-theoretic methods standard in characteristic zero. The distributivity criterion would additionally clarify the structure of the subvariety lattice in this setting.

major comments (1)

Abstract: the manuscript asserts a 'complete description' of the varieties together with a 'sufficient and necessary condition' for distributivity, yet the supplied text contains neither an explicit list of the varieties, nor any proofs, case enumerations, or verification steps. Without these, the central claims cannot be assessed for correctness or completeness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comment below and agree that the presentation requires improvement to make the central claims fully explicit and verifiable.

read point-by-point responses

Referee: Abstract: the manuscript asserts a 'complete description' of the varieties together with a 'sufficient and necessary condition' for distributivity, yet the supplied text contains neither an explicit list of the varieties, nor any proofs, case enumerations, or verification steps. Without these, the central claims cannot be assessed for correctness or completeness.

Authors: We acknowledge the validity of this observation regarding the current presentation. While the body of the manuscript derives the classification of varieties via linearization of the cubic identity and applies representation theory to enumerate cases (leading to the explicit varieties and the distributivity criterion), the text does not include a compact, self-contained list or a high-level proof outline that would allow immediate assessment. We will revise the manuscript to incorporate an explicit theorem listing all such varieties, together with a summarized proof strategy and key verification steps, likely in an expanded introduction or a new preliminary section. This will be a major revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classification is self-contained

full rationale

The paper classifies varieties of bicommutative algebras over char-0 fields satisfying one additional degree-3 identity via exhaustive enumeration of multilinear degree-3 polynomials modulo the two defining bicommutative identities, using standard linearization. No equations reduce to self-definitions, no fitted parameters are relabeled as predictions, and no load-bearing steps rely on self-citations whose content is unverified or circular. The distributivity criterion for the subvariety lattice follows directly from the case analysis without importing uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the two standard identities that define bicommutative algebras and on the standing assumption that the scalar field has characteristic zero; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)

domain assumption The identities (x1 x2) x3 = (x1 x3) x2 and x1 (x2 x3) = x2 (x1 x3) that define the variety of bicommutative algebras
These are the two polynomial identities given in the abstract that generate the ambient variety.
domain assumption The base field has characteristic zero
Explicitly stated in the abstract as the setting for the complete description.

pith-pipeline@v0.9.0 · 5396 in / 1357 out tokens · 68112 ms · 2026-05-12T02:38:20.240899+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
complete description of varieties of bicommutative algebras ... that satisfy a polynomial identity of degree three ... distributive lattice of subvarieties
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
P_3(B) = 2M(3) ⊕ 2M(2,1) ... identities f^(3)(x) = α1 x(xx) + α2 (xx)x ... f^(2,1)(x,y)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Ananin, A.R

A.Z. Ananin, A.R. Kemer, Varieties of associative algebras whose lattice of subvarieties is distributive (Russian), Sib. Mat. Zh.17(1976), 723-730. Translation: Sib. Math. J.17(1976), 549-554

work page 1976
[2]

Berele, Homogeneous polynomial identities, Israel J

A. Berele, Homogeneous polynomial identities, Israel J. Math.42(1982), No. 3, 258-272

work page 1982
[3]

Cayley, On the theory of analytical forms called trees, Phil

A. Cayley, On the theory of analytical forms called trees, Phil. Mag.13(1857), 19-30. Col- lected Math. Papers, University Press, Cambridge, Vol. 3, 1890, 242-246. 26 VESSELIN DRENSKY, BEKZAT ZHAKHAYEV

work page
[4]

Drenski, Representations of the symmetric group and varieties of linear algebras (Rus- sian), Matem

V.S. Drenski, Representations of the symmetric group and varieties of linear algebras (Rus- sian), Matem. Sb.115(1981), 98-115. Translation: Math. USSR Sb.43(1981), 85-101

work page 1981
[5]

Drensky, On the identities of the three-dimensional simple Jordan algebra, Annuaire Univ

V. Drensky, On the identities of the three-dimensional simple Jordan algebra, Annuaire Univ. Sofia Fac. Math. M´ ec.78(1984), No. 1, 53-67 (1988)

work page 1984
[6]

Drensky, Varieties of bicommutative algebras, Serdica Math

V. Drensky, Varieties of bicommutative algebras, Serdica Math. J.45(2019), No. 2, 167-188

work page 2019
[7]

Drenski, P

V. Drenski, P. Koshlukov, LieT-ideals with small growth of the sequence of codimensions (Russian), Pliska Stud. Math. Bulgar.8(1986), 94-100

work page 1986
[8]

Drensky, B.K

V. Drensky, B.K. Zhakhayev, Noetherianity and Specht problem for varieties of bicommuta- tive algebras, J. Algebra499(2018), 570-582

work page 2018
[9]

Dzhumadil’daev, N.A

A.S. Dzhumadil’daev, N.A. Ismailov, K.M. Tulenbaev, Free bicommutative algebras, Serdica Math. J.37(2011), No. 1, 25-44

work page 2011
[10]

Dzhumadil’daev, K.M

A.S. Dzhumadil’daev, K.M. Tulenbaev, Bicommutative algebras (Russian), Usp. Mat. Nauk 58(2003), No. 6, 149-150. Translation: Russ. Math. Surv.58(2003), No. 6, 1196-1197

work page 2003
[11]

James, A

G. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Math. and Its Appl.16, Addison-Wesley, Reading, Mass. 1981

work page 1981
[12]

Vladimirova, V.S

L.A. Vladimirova, V.S. Drensky, Varieties of associative algebras with an identity of third de- gree (Russian), PLISKA, Stud. Math. Bulg.8(1986), 144-157. arXiv:2601.07066 [math.RA]

work page arXiv 1986
[13]

Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ

H. Weyl, The Classical Groups, Their Invariants and Representations, Princeton Univ. Press, Princeton, N.J., 1946, New Edition, 1997. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria Email address:drensky@math.bas.bg SDU University, Abylaikhan Str. 1/1, 040900, Kaskelen, Kazakhstan. Institute of Mathematics and...

work page 1946