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A Frattini theory for evolution algebras

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Pith's one-line read This paper defines a canonical supersolvable nilradical for every finite-dimensional evolution algebra and uses it to characterize when the Frattini subalgebra and Frattini ideal are trivial, with applications to dually atomistic algebras.

desk verdict 'A solid, well-motivated first Frattini theory for evolution algebras, but the central existence theorem rests on a missing internal lemma ('Lemma 3.5') that must be supplied before the results can be accepted.' read the letter →

arxiv 2507.01935 v1 pith:2MKZTT5R submitted 2025-07-02 math.RA

Manuel Ladra , Andrés Pérez-Rodríguez This is my paper · ORCID
classification math.RA MSC 17D9217A6017B3006B05
keywords EvolutionalgebrasFrattinisubalgebraidealnilradicalsupersolvableduallyatomisticnon-associative
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

This paper aims to build a Frattini theory for evolution algebras, the non-associative algebras used to model non-Mendelian inheritance. Because more than one maximal nilpotent ideal can exist, the classical nilradical is not always defined; the authors instead construct a chain of $E$-supersolvable nilpotent ideals and define the supersolvable nilradical to be its terminal term. They then characterize, in several classes, exactly when the Frattini subalgebra and Frattini ideal vanish, and they classify the dually atomistic evolution algebras in two families. If the main existence theorem holds, every finite-dimensional evolution algebra has a canonical largest $E$-supersolvable nilpotent ideal, which would make the Frattini theory as robust as it is for Lie and Leibniz algebras.

What carries the argument

The load-bearing object is the $E$-supersolvable nilpotent series, a chain of ideals $0 \subseteq N^1(E) \subseteq N^2(E) \subseteq \cdots$ built inductively: $N^1(E)$ is the sum of the nilradicals of all basic ideals that lie in $T_K$ plus the annihilator, and each successive term $N^i(E)/N^{i-1}(E)$ is obtained by applying the same recipe to the quotient $E/N^{i-1}(E)$ (Definition 3.11). Its terminal term $N^r(E)$ is the supersolvable nilradical $\mathrm{SNil}(E)$. The paper's argument that this term is the largest $E$-supersolvable nilpotent ideal (Theorem 3.14) is what turns the construction into a canonical nilradical, and the later characterizations of the Frattini subalgebra, Frattini ideal, and dually atomistic algebras all reduce to computations of $\mathrm{SNil}(E)$.

What would settle it

Compute the $E$-supersolvable nilpotent series for a small algebra (for instance, an algebra built from two overlapping $T_K$ blocks, modifying Example 3.1) and check the projection condition invoked in Propositions 3.12–3.13: every element whose projection falls outside the terminal term must multiply some series generator $w_{i,j}$ into $K^* w_{i,j} + N^{i-1}(E)$. A single element violating this condition disproves the existence theorem for the supersolvable nilradical, collapsing the later classifications.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.14: for any finite-dimensional evolution algebra $E$, the terminal term $N^r(E)$ of the $E$-supersolvable nilpotent series is the largest $E$-supersolvable nilpotent ideal. The authors call this ideal the supersolvable nilradical, $\mathrm{SNil}(E)$, and use it to prove that in the family $T_K$ (solvable non-nilpotent algebras with one-dimensional derived subalgebra), the Frattini subalgebra and Frattini ideal vanish exactly when the algebra is isomorphic to $E_2(1,-1,0,\ldots,0)$ (Theorem 4.1). More generally, a $\phi$-free algebra must have basic nilradical equal to its annihilator (Theorem 4.3), with a converse under a support condition (Theorem 4.6). They also show that a dually atomistic almost-abelian algebra, or one whose supersolvable nilradical has full support, must be isomorphic to $E_2(1,-1)$ or $E_{n,1}$ (Theorem 5.3).

Load-bearing premise

The proof that the terminal term of the $E$-supersolvable nilpotent series is the largest $E$-supersolvable nilpotent ideal depends on an asserted structural fact about elements falling outside the series' terms—one that the paper invokes as 'Lemma 3.5' but never proves, and no such lemma appears in the text. If that fact is false, the supersolvable nilradical may not be well-defined and the later Frattini and duality results lose their foundation.

Editorial extensions

If this is right

  • If Theorem 3.14 stands, every finite-dimensional evolution algebra has a canonical largest $E$-supersolvable nilpotent ideal, giving a nilradical concept where the classical one fails.
  • In the family $T_K$, the Frattini subalgebra and the Frattini ideal coincide, and each is either zero or the whole derived subalgebra, depending only on whether the annihilator has codimension two.
  • A $\phi$-free evolution algebra must have basic nilradical equal to its annihilator; with the extra hypothesis that $\mathrm{SNil}(E)^2$ is an ideal, it must also satisfy $\mathrm{SNil}(E) = \mathrm{Asoc}_1(E)$.
  • Under the support condition $\operatorname{supp}(\mathrm{SNil}(E)) = \operatorname{supp}(E)$, being $\phi$-free is equivalent to splitting as a direct sum of copies of $E_2(1,-1)$ plus an abelian annihilator.
  • In the two families considered, being dually atomistic forces the algebra to be almost abelian and isomorphic to either $E_2(1,-1)$ or $E_{n,1}$.

Reading between the lines

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

  • The proof of Theorem 3.14 invokes a 'Lemma 3.5' that does not appear in the paper; until that structural projection statement is proved or replaced, the existence of the supersolvable nilradical as the largest such ideal should be treated as conditional.
  • If the missing lemma can be supplied, the construction suggests an algorithmic way to compute $\mathrm{SNil}(E)$ by repeatedly taking quotients and collecting nilradicals of $T_K$ blocks, which could be implemented for concrete evolution algebras.
  • The classification of dually atomistic algebras likely does not extend to all evolution algebras: the paper's own example of a dually atomistic algebra that is neither abelian, almost abelian, nor semisimple shows the Lie-algebra analogue fails here, and other exotic examples may exist outside the two families studied.
  • A testable extension is whether the triviality conditions in Theorem 4.6 remain equivalent when the support condition is dropped; Example 4.5 already shows the converse of Theorem 4.3 fails in general, so the support condition is likely essential.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a Frattini theory for finite-dimensional evolution algebras. It defines the Frattini subalgebra as the intersection of maximal subalgebras, the Frattini ideal as the largest ideal inside it, and introduces a canonical-looking "supersolvable nilradical" SNil(E) as the largest E-supersolvable nilpotent ideal. The construction proceeds through an inductively defined E-supersolvable nilpotent series N^i(E); Theorem 3.14 claims that the terminal term N^r(E) is indeed the largest such ideal. The later sections use SNil(E) to characterize when the Frattini subalgebra and ideal are trivial (Theorems 4.1, 4.3, 4.6) and to classify dually atomistic evolution algebras in certain families (Theorem 5.3).

Significance. If the main existence and uniqueness theorem is correct, the paper supplies a useful substitute for the classical nilradical in a setting where maximal nilpotent ideals need not be unique, and it connects this notion to Frattini theory in a way that yields concrete structural characterizations. The manuscript contains several explicit, verifiable computations (e.g., Example 3.16) and a strengthening of the classification of the family T_K in Proposition 2.6, which are valuable in themselves. However, the central construction currently rests on an unproved and apparently nonexistent lemma, so the claimed results are not yet verified as written.

major comments (3)
  1. [§3.2, Propositions 3.10, 3.12, 3.13] The proofs of Propositions 3.10, 3.12, and 3.13 repeatedly cite a "Lemma 3.5" for the load-bearing structural assertion, but no such lemma appears in the paper. The only nearby statement, Corollary 3.5, concerns ann_E(E^2)=Nil(E) for E in T_K and is not the assertion used. In particular, Proposition 3.10 uses "by Lemma 3.5" to conclude N^1(E)^{<3>}=0, and Propositions 3.12 and 3.13 use it to infer that an element outside N^k(E) multiplies some series generator w_{i,j} into K^* w_{i,j}+N^{i-1}(E). This missing statement is essential to the nilpotency and uniqueness arguments, so it must be stated and proved.
  2. [§3.2, Proposition 3.12] The necessity direction of Proposition 3.12 is incomplete: the assertion that if pi_{N^k(E)}(w) is not in N^k(E) then repeated multiplication by w produces a nonzero right-nilpotent series element depends entirely on the missing structural lemma. Without that lemma, the characterization of nilpotent ideals in terms of the E-supersolvable nilpotent series is not established.
  3. [§3.2, Proposition 3.13 and Theorem 3.14] Proposition 3.13, which shows that I+N^r(E) is nilpotent for every nilpotent ideal I, uses the same unproved projection property to force pi_{N^r(E)}(w) in N^r(E). Since Theorem 3.14 then uses Proposition 3.13 to prove that N^r(E) is the unique largest E-supersolvable nilpotent ideal, the definition of SNil(E) in Definition 3.15 and all later theorems depending on it are not yet supported. Supplying the missing lemma is therefore a necessary revision, not a cosmetic one.
minor comments (4)
  1. [§3.2, Definition 3.11] The chain in Definition 3.11 is written as "0 ⊆ N1(E) ⊆ ..." without superscripts; it should be N^1(E), N^2(E), and so on, for consistency with the surrounding notation.
  2. [§3.2, after equation (3.5)] The sentence "Since the nilradical of every E_{i,j} with j ∈ Γ1 is characterised" should refer to Γ_i rather than Γ_1.
  3. [§3.2, Proposition 3.10] In the nilpotency part of Proposition 3.10, the statement "N^{i+1}(E)^{<k>} ⊂ N^i(E)^{<k-2>} for any k ≥ 3" is asserted to follow from (3.4) but the argument is abbreviated; a few more details about products of w_{i+1,j} with elements of N^{i+1}(E) would improve readability.
  4. [§4, Theorem 4.6] In the proof of Theorem 4.6, the notation in Case (a) writes "u_1 u_k ∈ K^*(e_1+e_2) ⊂ U"; it would be clearer to explicitly identify the copy of K^* e_1+e_2 as Asoc_1(E) and to justify why it cannot lie in the complement U.

Circularity Check

1 steps flagged · score 1.0 of 10

No circular reduction found: the nilradical and Frattini results are derived from stated constructions and external theorems, though Theorem 3.14 depends on a missing Lemma 3.5.

  1. other [Section 3.2, Proposition 3.10 and Proposition 3.12 (also used in Proposition 3.13 and Theorem 3.14)]
    "Then, it holds that N1(E)⟨2⟩ = span {w1,j : j ∈ Λ1} and, by Lemma 3.5, N1(E)⟨3⟩ = 0, what yields the nilpotency of N1(E). ... if there exists an element w ∈ I such that πNk(E)(w) /∈ Nk(E), then, by Lemma 3.5, there exists an index i ≤ k and an element wi,j with j ∈ Γi such that wwi,j ∈ K∗wi,j + N i−1(E)."

    This is an omitted proof rather than a circular step. Lemma 3.5 is never stated; the only nearby label, Corollary 3.5, asserts annE(E2)=Nil(E) for E∈TK, which is not the assertion used. The missing projection property is load-bearing: Proposition 3.12 uses it for the necessity direction, Proposition 3.13 uses it to prove I+N^r(E) is nilpotent, and Theorem 3.14 uses those results to conclude N^r(E) is the largest E-supersolvable nilpotent ideal. The conclusion is therefore unverified as written, but it is not obtained by fitting, by defining the target into the premise, or by importing an unproved uniqueness result from the authors' prior work. It is a correctness gap, not circularity.

full rationale

The paper's derivation chain is largely self-contained. The supersolvable nilradical is introduced by an inductive construction (equations (3.3) and (3.5)) based on the proven classification of TK algebras in Theorem 3.4; Theorem 3.14 is a maximality theorem about that construction, not a re-labelling of an assumption. The Frattini results in Section 4 use standard external tools (Towers [22,23], Marshall [17]) and are not justified by self-citation. The only author-overlapping reference, [11], supplies the standard fact that nilpotent evolution algebras have strictly triangular structure matrices, which is independent support rather than a circular premise. However, the proof of Theorem 3.14 is incomplete: Propositions 3.10, 3.12 and 3.13 cite a nonexistent 'Lemma 3.5' for the key facts N^1(E)^{<3>}=0 and the projection property π_{N^k(E)}(w)∈N^k(E) forced by nilpotency. This is a serious correctness risk in the central existence/uniqueness claim, but it does not make the paper circular. Accordingly the circularity score is 1.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The paper introduces a new algebraic invariant, the supersolvable nilradical, whose construction depends on prior classification results and on a missing lemma. There are no fitted parameters or data. The main burden on the reader is accepting the structural property cited as 'Lemma 3.5'.

assumptions (4)
  • domain assumption Finite-dimensional evolution algebras over a field of characteristic not 2
    The paper restricts to char K ≠ 2 and finite-dimensional algebras throughout (Section 2.1).
  • domain assumption Classification of TK evolution algebras from Camacho et al. [10]
    Proposition 2.6 relies on [10, Proposition 2.5 & Remark 2.6] which characterizes solvable non-nilpotent evolution algebras with one-dimensional derived subalgebra as E_k(λ_1,...,λ_n).
  • ad hoc to paper Existence of the structural assertion in the missing 'Lemma 3.5'
    Propositions 3.12 and 3.13 invoke 'Lemma 3.5' to assert that an element outside the terminal term N^k(E) acts nontrivially on some generator w_{i,j} of the E-supersolvable nilpotent series. No such lemma appears in the paper; this is a load-bearing unproved assumption.
  • domain assumption Adapted Frattini theory results from Towers [22, 23]
    Lemmas 2.10, 2.11 and the non-generator characterization of F(E) are imported from [23] and [22] and used throughout Sections 4 and 5.
invented entities (1)
  • Supersolvable nilradical SNil(E)
    purpose: Largest E-supersolvable nilpotent ideal, a replacement for the classical nilradical which is not unique in evolution algebras.
    Introduced in Definition 3.15. Its existence is claimed in Theorem 3.14 but the proof depends on the unproved 'Lemma 3.5'. It has no falsifiable handle outside the paper; it is an internal algebraic construct.

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Pith. "Pith review of A Frattini theory for evolution algebras." pith.science (2026). https://pith.science/paper/2MKZTT5R

@misc{pith2026250701935,
  author       = {Pith},
  title        = {Pith review of: A Frattini theory for evolution algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2MKZTT5R}},
  note         = {Machine review of arXiv:2507.01935}
}
read the original abstract

This paper develops a Frattini theory for evolution algebras defining the Frattini subalgebra as the intersection of all maximal subalgebras, and the Frattini ideal as the largest ideal contained in it. To this end, we revisit the notion of nilradical, whose classical definition is not directly applicable in this setting, and propose the supersolvable nilradical as a suitable alternative. This leads to necessary and sufficient conditions for the triviality of the Frattini subalgebra and ideal. Finally, we also briefly examine the relevance of the Frattini ideal in the study of dually atomistic evolution algebras.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

24 extracted references · 22 canonical work pages

  1. [1]

    D. W. Barnes, On the cohomology of soluble Lie algebras, Math. Z. 101 (1967) 343–349. URL https://doi.org/10.1007/BF01109799

    work page doi:10.1007/bf01109799 1967

  2. [2]

    D. W. Barnes, H. M. Gastineau-Hills, On the theory of soluble Lie algebras, Math. Z. 106 (1968) 343–354. URL https://doi.org/10.1007/BF01115083

    work page doi:10.1007/bf01115083 1968

  3. [3]

    Batten, L

    C. Batten, L. Bosko-Dunbar, A. Hedges, J. T. Hird, K. Stagg, E. Stitzinger, A Frattini theory for Leibniz algebras, Comm. Algebra 41 (4) (2013) 1547–1557. URL https://doi.org/10.1080/00927872.2011.643844 A FRATTINI THEORY FOR EVOLUTION ALGEBRAS 21

    arXiv 2013

  4. [4]

    Bosko, A

    L. Bosko, A. Hedges, J. T. Hird, N. Schwartz, K. Stagg, Jacobson’s refinement of Engel’s theorem for Leibniz algebras, Involve 4 (3) (2011) 293–296. URL https://doi.org/10.2140/involve.2011.4.293

    work page doi:10.2140/involve.2011.4.293 2011

  5. [5]

    Boudi, Y

    N. Boudi, Y. Cabrera Casado, M. Siles Molina, Natural families in evolution algebras, Publ. Mat. 66 (1) (2022) 159–181. URL https://doi.org/10.5565/publmat6612206

    work page doi:10.5565/publmat6612206 2022

  6. [6]

    Cabrera Casado, Evolution algebras, Ph.D

    Y. Cabrera Casado, Evolution algebras, Ph.D. thesis, Universidad de M´ alaga, 2016

    work page 2016

  7. [7]

    Cabrera Casado, M

    Y. Cabrera Casado, M. Kanuni, M. Siles Molina, Basic ideals in evolution algebras, Linear Algebra Appl. 570 (2019) 148–180. URL https://doi.org/10.1016/j.laa.2019.01.010

    work page doi:10.1016/j.laa.2019.01.010 2019

  8. [8]

    Cabrera Casado, D

    Y. Cabrera Casado, D. Mart ´ ın Barquero, C. Mart ´ ın Gonz´ alez, A. Tocino, On simple evolution algebras of dimension two and three. Constructing simple and semisimple evolution algebras, Linear Multilinear Algebra 73 (3) (2025) 507–524. URL https://doi.org/10.1080/03081087.2024.2352452

    arXiv 2025

Show all 24 references
  1. [9]

    Cabrera Casado, M

    Y. Cabrera Casado, M. Siles Molina, M. V. Velasco, Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra Appl. 495 (2016) 122–162. URL https://doi.org/10.1016/j.laa.2016.01.007

    2016 doi

  2. [10]

    L. M. Camacho, J. R. G´ omez, B. A. Omirov, R. M. Turdibaev, Some properties of evolution algebras, Bull. Korean Math. Soc. 50 (5) (2013) 1481–1494. URL https://doi.org/10.4134/BKMS.2013.50.5.1481

    2013 doi

  3. [11]

    J. M. Casas, M. Ladra, B. A. Omirov, U. A. Rozikov, On evolution algebras, Algebra Colloq. 21 (2) (2014) 331–342. vol. 21, 2014. URL https://doi.org/10.1142/S1005386714000285

    2014 doi

  4. [12]

    Elduque, A

    A. Elduque, A. Labra, Evolution algebras and graphs, J. Algebra Appl. 14 (7) (2015) 1550103, 10. URL https://doi.org/10.1142/S0219498815501030

    2015 doi

  5. [13]

    Elduque, A

    A. Elduque, A. Labra, On nilpotent evolution algebras, Linear Algebra Appl. 505 (2016) 11–31. URL https://doi.org/10.1016/j.laa.2016.04.025

    2016 doi

  6. [14]

    Frattini, Intorno alla generazione dei gruppi di operazioni, Rom

    G. Frattini, Intorno alla generazione dei gruppi di operazioni, Rom. Acc. L. Rend. (4) I (1885) 281–285, 455–457

  7. [15]

    Hall, Jr., The theory of groups, Chelsea Publishing Co., New York, 1976, reprinting of the 1968 edition

    M. Hall, Jr., The theory of groups, Chelsea Publishing Co., New York, 1976, reprinting of the 1968 edition

    1976

  8. [16]

    Lincoln, D

    M. Lincoln, D. Towers, Frattini theory for restricted Lie algebras, Arch. Math. (Basel) 45 (5) (1985) 451–457. URL https://doi.org/10.1007/BF01195370

    1985 doi

  9. [17]

    E. I. Marshall, The Frattini subalgebra of a Lie algebra, J. London Math. Soc. 42 (1967) 416–422. URL https://doi.org/10.1112/jlms/s1-42.1.416

    1967 doi

  10. [18]

    P´ aez-Guill´ an, S

    P. P´ aez-Guill´ an, S. Siciliano, D. A. Towers, On the subalgebra lattice of a restricted Lie algebra, Linear Algebra Appl. 660 (2023) 47–65. URL https://doi.org/10.1016/j.laa.2022.12.004

    2023 doi

  11. [19]

    Scheiderer, Intersections of maximal subalgebras in Lie algebras, J

    C. Scheiderer, Intersections of maximal subalgebras in Lie algebras, J. Algebra 105 (1) (1987) 268–270. URL https://doi.org/10.1016/0021-8693(87)90192-X

    1987 doi

  12. [20]

    J. P. Tian, Evolution algebras and their applications, vol. 1921 of Lecture Notes in Mathematics, Springer, Berlin, 2008. URL https://doi.org/10.1007/978-3-540-74284-5

    1921 doi

  13. [21]

    J. P. Tian, P. Vojtˇ echovsk´ y, Mathematical concepts of evolution algebras in non-Mendelian genetics, Quasigroups Related Systems 14 (1) (2006) 111–122

    2006

  14. [22]

    D. A. Towers, On the generators of a nilpotent non-associative algebra, Quart. J. Math. Oxford Ser. (2) 22 (1971) 545–550. URL https://doi.org/10.1093/qmath/22.4.545 22 M. LADRA AND A. P ´EREZ-RODR´IGUEZ

    1971 doi

  15. [23]

    D. A. Towers, A Frattini theory for algebras, Proc. London Math. Soc. (3) 27 (1973) 440–462. URL https://doi.org/10.1112/plms/s3-27.3.440

    1973 doi

  16. [24]

    K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative, vol. 104 of Pure and Applied Mathematics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982, translated from the Russian by Harry F. Smith. 1Dep...

    1982

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