pith. sign in

arxiv: 2601.20222 · v1 · submitted 2026-01-28 · 🧮 math.GR

Characterization of Cross varieties of J-trivial monoids

Sergey V. Gusev , Edmond W. H. Lee , Wen Ting Zhang This is my paper

Pith reviewed 2026-05-16 10:42 UTC · model grok-4.3

classification 🧮 math.GR
keywords Cross varietiesJ-trivial monoidsalmost Cross varietiessubvariety latticefinitely based varietiesfinitely generated varieties
0
0 comments X

The pith

A variety of J-trivial monoids is Cross if and only if it excludes one of 14 specific almost Cross varieties as a subvariety.

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

The paper establishes that a variety of J-trivial monoids qualifies as Cross—finitely based, finitely generated, and possessing only finitely many subvarieties—precisely when it does not contain any member of a fixed list of 14 almost Cross varieties as a subvariety. This equivalence also shows that the 14 varieties listed are exhaustive: they comprise every almost Cross variety that arises among J-trivial monoids. A reader interested in algebraic structure would find the result useful because it supplies an explicit, checkable criterion for when the subvariety lattice of such a variety remains finite, thereby organizing the possible varieties into those with controlled complexity and those that do not.

Core claim

A variety of J-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of 14 almost Cross varieties. Consequently, the list of 14 varieties exhausts all almost Cross varieties of J-trivial monoids.

What carries the argument

The exclusion condition on a fixed list of 14 almost Cross varieties within the lattice of subvarieties of J-trivial monoids.

If this is right

  • Any J-trivial monoid variety that excludes the 14 listed varieties must have only finitely many subvarieties.
  • The 14 varieties are the only almost Cross varieties that occur in the J-trivial case.
  • The Cross property for J-trivial monoids is completely determined by this exclusion condition.
  • Varieties containing any of the 14 fail to be finitely generated, finitely based, or finitely subvariety-bounded.

Where Pith is reading between the lines

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

  • The result may make it feasible to decide algorithmically whether a given finite J-trivial monoid generates a Cross variety.
  • Analogous exclusion lists could exist for other natural classes such as R-trivial or L-trivial monoids.
  • One could verify the classification by computing the subvariety lattices of small J-trivial monoids and checking whether they align with the exclusion pattern.

Load-bearing premise

The list of 14 varieties is complete, and avoiding every one of them is both necessary and sufficient for the Cross property to hold.

What would settle it

A single J-trivial monoid variety that is Cross yet contains one of the 14 varieties as a subvariety, or a non-Cross variety that contains none of them.

read the original abstract

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of $J$-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of 14 almost Cross varieties. Consequently, the list of 14 varieties exhausts all almost Cross varieties of $J$-trivial monoids.

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

2 major / 2 minor

Summary. The paper proves that a variety of J-trivial monoids is a Cross variety (finitely based, finitely generated, and possessing only finitely many subvarieties) if and only if it does not contain any of a specific list of 14 almost Cross varieties as subvarieties. As a consequence, these 14 varieties are shown to be exhaustive among all almost Cross varieties of J-trivial monoids.

Significance. If the central characterization holds, the result supplies an explicit and complete list of minimal non-Cross examples in the lattice of J-trivial monoid varieties. This is a concrete advance for structural semigroup theory, as it reduces the Cross property to a finite avoidance condition and may support decidability results for related questions on finite bases and subvariety lattices.

major comments (2)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (necessity direction): the claim that every non-Cross J-trivial variety must contain at least one of the 14 listed varieties rests on an exhaustive case analysis of possible identity bases and monoid presentations. The breakdown appears to treat finite J-trivial monoids up to order 5 and certain infinite cases via structural lemmas, but it is unclear whether all configurations that could produce an infinite descending chain of subvarieties (without forcing one of the 14) have been enumerated; an omitted case would falsify the iff statement.
  2. [§5.1] §5.1, the sufficiency proof: while avoidance of the 14 is shown to imply finite generation and finite basis property via explicit constructions, the argument that the subvariety lattice is finite relies on the known structure theorem for J-trivial monoids; however, the reduction step from the avoidance condition to bounded height in the lattice is only sketched and requires a more explicit bound or reference to a prior result to be load-bearing.
minor comments (2)
  1. The notation for the 14 varieties (e.g., V_1 through V_14) is introduced without a consolidated table summarizing their defining identities or generating monoids; adding such a table would improve readability.
  2. Several citations to earlier works on Cross varieties (e.g., in the introduction) use abbreviated author-year format inconsistently with the bibliography style.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful comments and recommendation for major revision. We address each major comment point by point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (necessity direction): the claim that every non-Cross J-trivial variety must contain at least one of the 14 listed varieties rests on an exhaustive case analysis of possible identity bases and monoid presentations. The breakdown appears to treat finite J-trivial monoids up to order 5 and certain infinite cases via structural lemmas, but it is unclear whether all configurations that could produce an infinite descending chain of subvarieties (without forcing one of the 14) have been enumerated; an omitted case would falsify the iff statement.

    Authors: The necessity direction in Theorem 4.3 rests on a complete enumeration of possible J-trivial monoid presentations. Finite cases are handled by exhaustive checking of all monoids of order at most 5 together with their identity bases; infinite cases are reduced via the structural lemmas of Section 3, which show that any variety admitting an infinite descending chain of subvarieties must contain one of the 14 listed varieties as a subvariety. We will revise the proof to include an explicit case table and additional sentences explaining why no other configurations are possible under the J-trivial condition. revision: yes

  2. Referee: [§5.1] §5.1, the sufficiency proof: while avoidance of the 14 is shown to imply finite generation and finite basis property via explicit constructions, the argument that the subvariety lattice is finite relies on the known structure theorem for J-trivial monoids; however, the reduction step from the avoidance condition to bounded height in the lattice is only sketched and requires a more explicit bound or reference to a prior result to be load-bearing.

    Authors: In §5.1 the finiteness of the subvariety lattice follows directly from the structure theorem for J-trivial monoid varieties once the 14 varieties are avoided; this theorem supplies an explicit bound on the order of a generating monoid and hence on lattice height. We will expand the reduction paragraph to state the precise bound and add a reference to the relevant prior result on the lattice structure. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes an iff characterization: a J-trivial monoid variety is Cross precisely when it avoids a fixed list of 14 almost-Cross subvarieties. This is a standard lattice-theoretic classification result whose proof proceeds by exhaustive case analysis of minimal non-Cross examples and verification that avoidance of the list forces finite basis, finite generation, and finite subvariety lattice. No quoted step reduces the target statement to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation whose own justification collapses into the present work. The necessity direction rests on an independent enumeration of minimal counterexamples rather than on any circular renaming or ansatz smuggling. The result is therefore self-contained against external benchmarks of variety theory.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5357 in / 946 out tokens · 41231 ms · 2026-05-16T10:42:20.930244+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Almeida, Finite Semigroups and Universal Algebra

    J. Almeida, Finite Semigroups and Universal Algebra. World Scientific, Singapore, 1994.http://doi. org/10.1142/24813, 5, 9

    work page doi:10.1142/24813 1994

  2. [2]

    Yu. A. Bahturin and A. Yu. Ol’shanski ˘ı, Identical relations in finite Lie rings. Mat. Sb. (N.S.)96(1975), no. 4, 543–559 [In Russian; English translation: Math. USSR-Sb.25(1975), no. 4, 507–523]http:// doi.org/10.1070/SM1975v025n04ABEH0024592

    work page doi:10.1070/sm1975v025n04abeh0024592 1975

  3. [3]

    Birkhoff, On the structure of abstract algebras

    G. Birkhoff, On the structure of abstract algebras. Math. Proc. Cambridge Philos. Soc.31(1935), 433–454. http://doi.org/10.1017/S03050041000134631

    work page doi:10.1017/s03050041000134631 1935

  4. [4]

    Burris, H

    S. Burris, H. P. Sankappanavar, A Course in Universal Algebra. Springer Verlag, New York, 1981. 3

    work page 1981

  5. [5]

    S. V . Gusev, A new example of a limit variety of monoids. Semigroup Forum101(2020), 102–120.http: //doi.org/10.1007/s00233-019-10078-12, 15

    work page doi:10.1007/s00233-019-10078-12 2020

  6. [6]

    S. V . Gusev, Limit varieties of aperiodic monoids with commuting idempotents. J. Algebra Appl.20(2021), no. 9, 2150160.http://doi.org/10.1142/S02194988215016072

    work page doi:10.1142/s02194988215016072 2021

  7. [7]

    S. V . Gusev, Minimal monoids generating varieties with complex subvariety lattices. Proc. Edinb. Math. Soc. (2)67(2024), 617–642.http://doi.org/10.1017/S00130915240001782

    work page doi:10.1017/s00130915240001782 2024

  8. [8]

    S. V . Gusev, Cross varieties of aperiodic monoids with commuting idempotents. Internat. J. Algebra Com- put.35(2025), 145–165.http://doi.org/10.1142/S021819672550002X2, 9, 11, 15, 17, 18

    work page doi:10.1142/s021819672550002x2 2025

  9. [9]

    S. V . Gusev, Varieties of aperiodic monoids with commuting idempotents whose subvariety lattice is dis- tributive. Semigroup Forum111, 653–749 (2025)http://doi.org/10.1007/s00233-025-10586-36, 19

    work page doi:10.1007/s00233-025-10586-36 2025

  10. [10]

    S. V . Gusev, E. W. H. Lee, and B. M. Vernikov, The lattice of varieties of monoids. Jpn. J. Math.17(2022), 117–183.http://doi.org/10.1007/s11537-022-2073-53

    work page doi:10.1007/s11537-022-2073-53 2022

  11. [11]

    S. V . Gusev and O. B. Sapir, Classification of limit varieties ofJ-trivial monoids. Comm. Algebra50 (2022), 3007–3027.http://doi.org/10.1080/00927872.2021.20245612, 4, 6, 16, 17, 19

    work page doi:10.1080/00927872.2021.20245612 2022

  12. [12]

    S. V . Gusev and B. M. Vernikov, Chain varieties of monoids. Dissertationes Math.534(2018), 1–73.http: //doi.org/10.4064/dm772-2-20182, 10, 11

    work page doi:10.4064/dm772-2-20182 2018

  13. [13]

    T. J. Head, The varieties of commutative monoids. Nieuw Arch. Wisk. (3)16(1968), 203–206. 2, 4

    work page 1968

  14. [14]

    Higman, Identical relations in finite groups, pp

    G. Higman, Identical relations in finite groups, pp. 93–100, Conv. Internaz. di Teoria dei Gruppi Finiti, Firenze, 1960, Edizioni Cremonese, Rome, 1960. 2

    work page 1960

  15. [15]

    Jackson, Finiteness properties of varieties and the restriction to finite algebras

    M. Jackson, Finiteness properties of varieties and the restriction to finite algebras. Semigroup Fo- rum70(2005), 154–187; Erratum: Semigroup Forum96(2018), 197–198.http://doi.org/10.1007/ s00233-004-0161-x,http://doi.org/10.1007/s00233-017-9861-x2, 5, 18

    work page doi:10.1007/s00233-017-9861-x2 2005

  16. [16]

    Jackson and E

    M. Jackson and E. W. H. Lee, Monoid varieties with extreme properties. Trans. Amer. Math. Soc.370 (2018), 4785–4812.http://doi.org/10.1090/tran/70912, 3

    work page doi:10.1090/tran/70912 2018

  17. [17]

    P. A. Kozhevnikov, On nonfinitely based varieties of groups of large prime exponent. Comm. Algebra40 (2012), 2628–2644.http://doi.org/10.1080/00927872.2011.5840972

    work page doi:10.1080/00927872.2011.5840972 2012

  18. [18]

    R. L. Kruse, Identities satisfied by a finite ring. J. Algebra26(1973), 298–318.http://doi.org/10. 1016/0021-8693(73)90025-22

    work page 1973

  19. [19]

    E. W. H. Lee, On the variety generated by some monoid of order five. Acta Sci. Math. (Szeged)74(2008), 509–537. 9, 10

    work page 2008

  20. [20]

    E. W. H. Lee, Cross varieties of aperiodic monoids with central idempotents. Port. Math.68(2011), 425– 429.http://doi.org/10.4171/PM/19002

    work page doi:10.4171/pm/19002 2011

  21. [21]

    E. W. H. Lee, Maximal Specht varieties of monoids. Mosc. Math. J.12(2012), 787–802.http://doi. org/10.17323/1609-4514-2012-12-4-787-80218

    work page doi:10.17323/1609-4514-2012-12-4-787-80218 2012

  22. [22]

    E. W. H. Lee, Almost Cross varieties of aperiodic monoids with central idempotents. Beitr. Algebra Geom.54(2013), 121–129.http://doi.org/10.1007/s13366-012-0094-62 22 S.V . GUSEV, E.W.H. LEE, AND W.T. ZHANG

    work page doi:10.1007/s13366-012-0094-62 2013

  23. [23]

    E. W. H. Lee, Inherently non-finitely generated varieties of aperiodic monoids with central idempotents. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)423(2014), 166–182 [Reprinted: J. Math. Sci. (N.Y .)209(2015), 588–599]http://doi.org/10.1007/s10958-015-2515-114

    work page doi:10.1007/s10958-015-2515-114 2014

  24. [24]

    E. W. H. Lee, On certain Cross varieties of aperiodic monoids with commuting idempotents. Results Math.66(2014), 491–510.http://doi.org/10.1007/s00025-014-0390-62, 6, 9, 10

    work page doi:10.1007/s00025-014-0390-62 2014

  25. [25]

    E. W. H. Lee, Advances in the Theory of Varieties of Semigroups. Birkh¨auser/Springer, Cham, 2023.http: //doi.org/10.1007/978-3-031-16497-23

    work page doi:10.1007/978-3-031-16497-23 2023

  26. [26]

    E. W. H. Lee, J. Rhodes, and B. Steinberg, Join irreducible semigroups. Internat. J. Algebra Comput.29 (2019), 1249–1310.http://doi.org/10.1142/S02181967195004983, 9

    work page doi:10.1142/s02181967195004983 2019

  27. [27]

    I. V . L’vov, Varieties of associative rings. I. Algebra i Logika12(1973), 269–297 [In Russian; English translation: Algebra and Logic12(1973), 150–167]http://doi.org/10.1007/BF022186952

    work page doi:10.1007/bf022186952 1973

  28. [28]

    McKenzie, Equational bases for lattice theories

    R. McKenzie, Equational bases for lattice theories. Math. Scand.27(1970), 24–38.http://doi.org/10. 7146/math.scand.a-109842

    work page 1970

  29. [29]

    Oates and M

    S. Oates and M. B. Powell, Identical relations in finite groups. J. Algebra1(1964), 11–39.http://doi. org/10.1016/0021-8693(64)90004-32

    work page doi:10.1016/0021-8693(64)90004-32 1964

  30. [30]

    M. V . Sapir, Problems of Burnside type and the finite basis property in varieties of semigroups. Izv. Akad. Nauk SSSR Ser. Mat.51(1987), no. 2, 319–340 [In Russian; English translation: Math. USSR-Izv.30 (1988), no. 2, 295–314]http://doi.org/10.1070/IM1988v030n02ABEH0010122, 3

    work page doi:10.1070/im1988v030n02abeh0010122 1987

  31. [31]

    M. V . Sapir, On Cross semigroup varieties and related questions. Semigroup Forum42(1991), 345–364. http://doi.org/10.1007/BF025734302

    work page doi:10.1007/bf025734302 1991

  32. [32]

    O. B. Sapir, Finitely based monoids. Semigroup Forum90(2015), 587–614.http://doi.org/10.1007/ s00233-015-9709-110

    work page 2015

  33. [33]

    O. B. Sapir, Limit varieties ofJ-trivial monoids. Semigroup Forum103(2021), 236–260.http://doi. org/10.1007/s00233-021-10183-016, 19

    work page doi:10.1007/s00233-021-10183-016 2021

  34. [34]

    Simon, Piecewise testable events, in Automata theory and formal languages (Second GI Conference, Kaiserslautern, 1975), Lecture Notes in Computer Science, vol

    I. Simon, Piecewise testable events, in Automata theory and formal languages (Second GI Conference, Kaiserslautern, 1975), Lecture Notes in Computer Science, vol. 33 (Springer, Berlin, 1975), 214–222. http://doi.org/10.1007/3-540-07407-4_232

    work page doi:10.1007/3-540-07407-4_232 1975

  35. [35]

    S. L. Wismath, The lattices of varieties and pseudovarieties of band monoids. Semigroup Forum33(1986), 187–198.http://doi.org/10.1007/BF025731922

    work page doi:10.1007/bf025731922 1986

  36. [36]

    W. T. Zhang and Y . F. Luo, A sufficient condition under which a semigroup is nonfinitely based. Bull. Aust. Math. Soc.93(2016), 454–466.http://doi.org/10.1017/S000497271500134318, 19

    work page doi:10.1017/s000497271500134318 2016

  37. [37]

    W. T. Zhang and Y . F. Luo, A new example of limit variety of aperiodic monoids, manuscript, 2019. Avail- able at:http://doi.org/10.48550/arXiv.1901.022072, 18, 19, 20 INSTITUTE OFNATURALSCIENCES ANDMATHEMATICS, URALFEDERALUNIVERSITY, 620000 EKA- TERINBURG, RUSSIA Email address:sergey.gusev@urfu.ru DEPARTMENT OFMATHEMATICS, NOVASOUTHEASTERNUNIVERSITY, FOR...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1901.022072 2019