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arxiv: 2312.01770 · v2 · submitted 2023-12-04 · 🧮 math.GR

The finite basis problem for endomorphism semirings of finite chains

Sergey V. Gusev , Mikhail V. Volkov This is my paper

Pith reviewed 2026-05-24 04:56 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite basis problemendomorphism semiringssemilatticeschainsidentitiessemirings
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The pith

The endomorphism semiring of the 3-element chain has no finite identity basis.

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

The paper shows that the set of all order-preserving maps on a three-element chain, made into a semiring by pointwise addition and function composition, satisfies no finite collection of identities that fully describe its equational theory. Combined with earlier results on finite semilattices with zero, this settles the finite-basis question for every endomorphism semiring arising from a finite chain. A reader would care because the result determines precisely which of these natural algebraic structures on ordered sets can be captured by finitely many equations.

Core claim

We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka, gives a complete solution to the finite basis problem for semirings of the form End(S) where S is a finite chain.

What carries the argument

The endomorphism semiring End(S) of the 3-element chain S, with pointwise addition and composition as operations.

If this is right

  • The finite basis problem is now completely solved for every finite chain.
  • The 3-element case is the smallest chain whose endomorphism semiring requires infinitely many identities.
  • Any chain with three or more elements yields an endomorphism semiring without a finite identity basis.

Where Pith is reading between the lines

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

  • The same non-finite-basis phenomenon may appear in endomorphism semirings of other small posets beyond chains.
  • The construction supplies concrete infinite families of identities that any purported finite basis would have to miss.

Load-bearing premise

The 3-element chain is treated as a semilattice under the standard order, with endomorphisms as order-preserving maps and semiring operations as pointwise addition plus composition.

What would settle it

Either an explicit finite list of identities satisfied by the semiring whose consequences include every identity true in it, or an infinite family of identities shown to be independent over the semiring.

read the original abstract

For every semilattice $\mathcal{S}=(S,+)$, the set $\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the 3-element chain has no finite identity basis. This, combined with earlier results by Dolinka (The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61, 441-448 (2009)), gives a complete solution to the finite basis problem for semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a finite chain.

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

0 major / 2 minor

Summary. The manuscript proves that the endomorphism semiring of the 3-element chain has no finite identity basis, by exhibiting an infinite independent set of identities satisfied by this 10-element semiring. Combined with Dolinka (2009), this yields a complete solution to the finite basis problem for End(S) where S is any finite chain.

Significance. If correct, the result completes the classification of finite basis properties for these semirings, a concrete advance in the finite basis problem for semiring varieties. The technique of constructing an explicit infinite independent family of identities is a standard and falsifiable method in universal algebra.

minor comments (2)
  1. [Abstract] Abstract: the parenthetical reference to the 10-element semiring (mentioned in the body) could be added for immediate context on the size of the algebra.
  2. The dependence on Dolinka (2009) for the remaining cases is appropriate, but the reference list should include the full bibliographic details of that paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its significance in completing the finite basis problem for endomorphism semirings of finite chains, and the recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring response or revision at this time.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes its central result—that End(S) for the 3-element chain has no finite identity basis—by directly exhibiting an infinite independent family of identities satisfied by this specific 10-element semiring. This is a standard mathematical proof technique relying on the explicit semiring structure (pointwise addition and composition of order-preserving maps on the chain semilattice). The combination with Dolinka (2009) is an external citation to a different author and does not form a self-citation chain. No parameters are fitted and renamed as predictions, no ansatz is smuggled via self-citation, and no uniqueness theorem is imported from the authors' prior work. The modeling assumptions are the conventional definitions for endomorphism semirings and do not reduce the claim to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; ledger entries are therefore minimal and drawn from standard background.

axioms (1)
  • standard math Endomorphisms of a semilattice form a semiring under pointwise addition and composition
    Invoked in the first sentence of the abstract as the setup for the problem.

pith-pipeline@v0.9.0 · 5641 in / 1075 out tokens · 20284 ms · 2026-05-24T04:56:44.913347+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. A nonfinitely based additively idempotent semiring of order four

    math.GR 2026-05 unverdicted novelty 4.0

    A 4-element additively idempotent semiring whose additive reduct has two minimal elements and two coatoms has no finite basis for its identities.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    Birkhoff, G.: Lattice Theory, American Mathematical So ciety, Providence, RI (1967)

    work page 1967

  2. [2]

    Springer, Berlin, Heidelberg, New Y ork (1981)

    Burris, S., Sankappanavar, H.P .: A Course in Universal A lgebra. Springer, Berlin, Heidelberg, New Y ork (1981)

    work page 1981

  3. [3]

    Chen, Y uzhu, Hu, Xun, Kitov, N.V ., Luo, Y anfeng, V olkov,M.V .: Identities of the Kauffman monoid K3. Comm. Algebra 48, 1956–1968 (2020)

    work page 1956

  4. [4]

    I, American Mathematical Society, Providence, RI (1961)

    Clifford A.H., Preston G.B.: The Algebraic Theory of Sem igroups, V ol. I, American Mathematical Society, Providence, RI (1961)

    work page 1961

  5. [5]

    Cohn, P .M.: Universal Algebra, Mathematics and Its Appl ications, vol. 6. D. Reidel Publishing Company, Dordrecht (1981)

    work page 1981

  6. [6]

    Algebra Universalis 60, 19–35 (2009)

    Dolinka, I.: A class of inherently nonfinitely based semi rings. Algebra Universalis 60, 19–35 (2009)

    work page 2009

  7. [7]

    Algebra Universalis 61, 441–448 (2009)

    Dolinka, I.: The finite basis problem for endomorphism se mirings of finite semilattices with zero. Algebra Universalis 61, 441–448 (2009)

    work page 2009

  8. [8]

    Zhao, X.Z.: V arieties generated b y ordered bands I

    Ghosh, S., Pastijn, F. Zhao, X.Z.: V arieties generated b y ordered bands I. Order 22, 109–128 (2005)

    work page 2005

  9. [9]

    Semigroup Forum 106, 403–420 (2023)

    Gusev, S.V ., V olkov, M.V .: Semiring identities of finiteinverse semigroups. Semigroup Forum 106, 403–420 (2023)

    work page 2023

  10. [10]

    Gusev, S.V ., V olkov, M.V .: Semiring and involution identities of power groups. J. Austral. Math. Soc. 115, 354–374 (2023)

    work page 2023

  11. [11]

    Clarend on Press, Oxford (1995)

    Howie J.M.: Fundamentals of Semigroup Theory. Clarend on Press, Oxford (1995)

    work page 1995

  12. [12]

    Algebra 611, 211–245 (2022)

    Jackson, M., Ren, Miaomiao, Zhao, Xianzhong: Nonfinite ly based ai-semirings with finitely based semi- group reducts, J. Algebra 611, 211–245 (2022)

    work page 2022

  13. [13]

    Semigroup Forum 78, 21–26 (2009)

    Jeˇ zek, J., Kepka, T., Mar´ oti, M.: The endomorphism semiring of a semilattice. Semigroup Forum 78, 21–26 (2009)

    work page 2009

  14. [14]

    Jones, P .R.: The semigroups B2 and B0 are inherently nonfinitely based, as restriction semigroup s. Int. J. Algebra Comput. 23, 1289–1335 (2013)

    work page 2013

  15. [15]

    Jones, P .R.: V arieties of left restriction semigroups. J. Aust. Math. Soc. 105, 173–200 (2018)

    work page 2018

  16. [16]

    Kad ’ ourek, J.: On varieties of combinatorial inverse s emigroups. I. Semigroup Forum 43, 305–330 (1991)

    work page 1991

  17. [17]

    Semigroup Forum 67, 317–343 (2003)

    Kad ’ ourek, J.: On bases of identities of finite inverse semigroups with solvable subgroups. Semigroup Forum 67, 317–343 (2003)

    work page 2003

  18. [18]

    Semigroup Forum 71, 27–48 (2005)

    Kuˇ ril, M., Pol´ ak, L.: On varieties of semilattice-ordered semigroups. Semigroup Forum 71, 27–48 (2005)

    work page 2005

  19. [19]

    The Theory of Partia l Symmetries

    Lawson, M.V .: Inverse Semigroups. The Theory of Partia l Symmetries. World Scientific, Singapore (1999)

    work page 1999

  20. [20]

    McAlister, D.B.: Semilattice ordered inverse semigro ups. In J.M. Andr´ e et al (eds.), Semigroups and Formal Languages, pp. 205–218. World Scientific, New Jersey (2007)

    work page 2007

  21. [21]

    McNulty, G.F., Sz´ ekely, Z., Willard, R.: Equational c omplexity of the finite algebra membership problem. Int. J. Algebra Comput. 18, 1283–1319 (2008)

    work page 2008

  22. [22]

    John Wiley & Sons, New Y ork (1984) 24 S

    Petrich, M.: Inverse Semigroups. John Wiley & Sons, New Y ork (1984) 24 S. V . GUSEV AND M. V . VOLKOV

    work page 1984

  23. [23]

    Sapir, M.V .: Problems of Burnside type and the finite bas is property in varieties of semigroups. Izv. Akad. Nauk SSSR, Ser. Mat. 51, 319–340 (1987) [In Russian; English translation: Mathema tics of the USSR–Izv. 30, 295–314 (1988)]

    work page 1987

  24. [24]

    Sapir, M.V .: Inherently nonfinitely based finite semigr oups. Mat. Sb. 133(2), 154–166 (1987) [In Russian; English translation: Mathematics of the USSR–Sb. 61, 155–166 (1988)]

    work page 1987

  25. [25]

    Czechoslovak Math

    Schein, B.M.: Completions, translational hulls and id eal extensions of inverse semigroups. Czechoslovak Math. J. 23(4), 575–610 (1973)

    work page 1973

  26. [26]

    Semigroup Forum 72, 207–222 (2006)

    Seif, S., Szab´ o, Cs.: Computational complexity of che cking identities in 0-simple semigroups and matrix semigroups over finite fields. Semigroup Forum 72, 207–222 (2006)

    work page 2006

  27. [27]

    Shaheen, S., Willard, R.: Algebras from finite group act ions and a question of Eilenberg and Sch¨ utzenberger. J. Algebra 631, 896–910 (2023)

    work page 2023

  28. [28]

    Algebra Universalis 82, Article no

    V olkov, M.V .: Semiring identities of the Brandt monoid. Algebra Universalis 82, Article no. 42 (2021)

    work page 2021