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arxiv: 1907.05953 · v1 · pith:SIJM6JDDnew · submitted 2019-07-11 · 🧮 math.RA

Word problem for finitely presented metabelian Poisson algebras

Zerui Zhang , Yuqun Chen , L.A. Bokut This is my paper

Pith reviewed 2026-05-24 23:02 UTC · model grok-4.3

classification 🧮 math.RA
keywords metabelian Poisson algebrasword problemGröbner-Shirshov baseslinear basisfinitely presented algebrassolvable word problemrewriting systems
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The pith

The word problem for finitely presented metabelian Poisson algebras is solvable.

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

The paper constructs an explicit linear basis for the free metabelian Poisson algebra on any well-ordered generating set, with the basis depending on the characteristic of the ground field. It adapts the Gröbner-Shirshov basis technique to this setting and uses the resulting rewriting systems to prove that equality of elements can be decided algorithmically in every finitely presented quotient. A reader cares because the word problem asks whether there exists a uniform procedure to determine when two formal expressions represent the same element, a foundational decidability question for any variety of algebras. If the claim holds, every finite set of generators and relations in this class yields an algebra whose elements admit effective normal forms.

Core claim

We first construct a linear basis for a free metabelian Poisson algebra generated by an arbitrary well-ordered set. It turns out that such a linear basis depends on the characteristic of the underlying field. Then we elaborate the method of Gröbner--Shirshov bases for metabelian Poisson algebras. Finally, we show that the word problem for finitely presented metabelian Poisson algebras are solvable.

What carries the argument

The Gröbner-Shirshov basis method for metabelian Poisson algebras, built on a characteristic-dependent linear basis for the free algebra that produces terminating, confluent rewriting systems in all finitely presented quotients.

If this is right

  • Any two elements given by words can be compared by reducing both to normal form via the rewriting system.
  • Finitely presented metabelian Poisson algebras admit an effective procedure for deciding membership in the ideal generated by the relations.
  • The decision procedure works uniformly whether the base field has characteristic zero or positive characteristic, using the appropriate variant of the basis.

Where Pith is reading between the lines

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

  • The same basis and rewriting technique may extend to algorithmic questions such as computing the center or derived series in these algebras.
  • Results of this type often serve as a stepping stone toward deciding the isomorphism problem for finitely presented objects in the variety.
  • One could implement the rewriting system for small presentations to obtain concrete normal forms and test growth rates of the algebras.

Load-bearing premise

The linear basis constructed for the free metabelian Poisson algebra is complete enough to support a terminating Gröbner-Shirshov rewriting system in every finitely presented quotient.

What would settle it

An explicit finite presentation of a metabelian Poisson algebra together with two distinct words that the rewriting system reduces to the same normal form but that are not equal in the algebra, or two equal words that reduce to distinct normal forms.

read the original abstract

We first construct a linear basis for a free metabelian Poisson algebra generated by an arbitrary well-ordered set. It turns out that such a linear basis depends on the characteristic of the underlying field. Then we elaborate the method of Gr\"{o}bner--Shirshov bases for metabelian Poisson algebras. Finally, we show that the word problem for finitely presented metabelian Poisson algebras are solvable.

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 paper first constructs an explicit linear basis for the free metabelian Poisson algebra on a well-ordered generating set, with separate cases according to the characteristic of the base field. It then develops the Gröbner-Shirshov basis technique adapted to metabelian Poisson algebras and applies the resulting rewriting system to arbitrary finitely presented quotients, proving termination and confluence on the constructed basis and thereby obtaining an algorithm that reduces every word to a unique normal form.

Significance. If the central claims hold, the work supplies a concrete decision procedure for the word problem in the variety of metabelian Poisson algebras. The explicit, characteristic-dependent basis construction together with the verification that the Gröbner-Shirshov basis remains confluent after adjoining arbitrary relations constitutes a standard but technically nontrivial extension of known results for Lie and associative algebras; the reproducibility of the normal-form algorithm is a clear strength.

minor comments (2)
  1. Abstract, final sentence: the subject-verb agreement error ('the word problem ... are solvable') should be corrected to 'is solvable'.
  2. The manuscript would benefit from an explicit statement, early in the introduction, of the precise monomial order employed on the constructed basis, since this order is load-bearing for the termination argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept the manuscript. The report accurately summarizes the main results on the linear basis for free metabelian Poisson algebras (characteristic-dependent) and the application of Gröbner-Shirshov bases to solve the word problem in finitely presented quotients.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds by explicit construction of a characteristic-dependent linear basis for the free metabelian Poisson algebra, followed by definition of a monomial order and a Gröbner-Shirshov basis for the metabelian identities plus arbitrary relations. Termination and confluence are shown directly on this basis, yielding a normal-form algorithm for finitely presented quotients. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or definitional renaming; the argument is self-contained against the constructed objects and does not rely on load-bearing prior results by the same authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard algebraic axioms for Poisson algebras and fields of arbitrary characteristic; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (2)
  • domain assumption The underlying field may have arbitrary characteristic
    The linear basis is stated to depend on characteristic, so the proof must handle both zero and positive cases.
  • standard math Gröbner-Shirshov bases exist and terminate for the constructed rewriting system
    The method is elaborated from prior Gröbner-Shirshov theory rather than proved from scratch in the abstract.

pith-pipeline@v0.9.0 · 5588 in / 1262 out tokens · 19873 ms · 2026-05-24T23:02:37.567793+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    A. T. Abdykhalykov, A. A. Mikhalev and U. U. Umirbaev, Aut omorphisms of two- generated free Leibniz algebras, Communication in Algebra , 29(7) (2001) 2953–2960

    work page 2001

  2. [2]

    A. L. Agore and G. Militaru, The global extension problem , crossed products and co-flag non-commutative Poisson algebras, J. Algebra 426 (2 015) 1–31

    work page

  3. [3]

    Becker, V

    T. Becker, V. Weispfenning, H. Kredel, Gr¨ obner bases: a computational approach to commutative algebra, Graduate Texts in Mathematics, 1993

    work page 1993

  4. [4]

    L. A. Bokut, A basis for free polynilpotent Lie algebras, Algebra i Logika Sem. 2(4) (1963) 13–19

    work page 1963

  5. [5]

    Buchberger, An algorithm for finding the basis element s of the residue class ring of a zero dimensional polynomial ideal, Translated from the 1965 German original by Michael P

    B. Buchberger, An algorithm for finding the basis element s of the residue class ring of a zero dimensional polynomial ideal, Translated from the 1965 German original by Michael P. Abramson, J. Symbolic Comput. 41 (2006) 475–511

    work page 1965

  6. [6]

    Y. S. Chen, Y. Q. Chen, Gr¨ obner-Shirshov bases for metab elian Lie algebras, Journal of Algebra, 358(2012) 143–161

    work page 2012

  7. [7]

    Hall, Finiteness conditions for soluble groups, Proc

    P. Hall, Finiteness conditions for soluble groups, Proc . London Math. Soc. (Series 3), 4 (1954), 419–436

    work page 1954

  8. [8]

    Hilbert, Ueber die Theorie der algebraischen Formen, Mathematische Annalen, 36(4) (1890) 473–534

    D. Hilbert, Ueber die Theorie der algebraischen Formen, Mathematische Annalen, 36(4) (1890) 473–534. 26 ZERUI ZHANG, YUQUN CHEN ♯, AND LEONID A. BOKUT †

    work page

  9. [9]

    O. G. Kharlampovich, M.V. Sapir, Algorithmic problems i n varieties, International Journal of Algebra and Computation, 5(4) (1995) 379–602

    work page 1995

  10. [10]

    A. I. Shirshov, Some algorithmic problems for Lie algeb ras, Sibirsk Mat. Zh. 3(2) (1962) 292–296

    work page 1962

  11. [11]

    U. U. Umirbaev, Equality problem for center-by-metabe lian Lie algebras, translated from Algebra i Logika, 23(3) (1984) 305–318

    work page 1984

  12. [12]

    Zhang, Y

    Z. Zhang, Y. Q. Chen and L.A. Bokut, Some algorithmic pro blems for Poisson algebras, Journal of Algebra, 525 (2019) 562–588. Z.Z., School of Mathematical Sciences, South China Normal U niversity, Guangzhou 510631, P. R. China E-mail address : 295841340@qq.com Y.C., School of Mathematical Sciences, South China Normal U niversity, Guangzhou 510631, P. R. C...

    work page 2019