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A Normal form for HNN-extension of Dialgebras

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Pith's one-line read An explicit Groebner–Shirshov basis and normal form for HNN-extensions of dialgebras

desk verdict The Gröbner–Shirshov set S in this paper drops the derivation term d(a) from the HNN relation and kills all base products, so the normal form in Theorem 3.1 is for a different quotient, not for D*_d. read the letter →

arxiv 1908.08397 v2 pith:MBTLMOB3 submitted 2019-08-19 math.RA

Chia Zargeh This is my paper · ORCID
classification math.RA MSC 17A3617A5017A9913P1016S15
keywords normalformdialgebradiassociativealgebraGroebner-ShirshovbasisHNN-extensionComposition-Diamondlemmaderivation
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 tries to establish an explicit Groebner–Shirshov basis for the HNN-extension of a dialgebra and, as a consequence, a normal form for every element of the quotient. The intended payoff is a linear basis for the extension and an embedding of the original dialgebra into it, the dialgebra analogue of the classical HNN embedding theorem. The argument rests on a strengthened Composition–Diamond lemma in which a set of polynomials is a Groebner–Shirshov basis exactly when its irreducible diwords form a basis. On a sympathetic reading, the normal form is established for the presentation in which the derivation term $d(a)$ is zero; the set $S$ used in the paper contains only $[xx]_1$, $[xx]_2$, $[xy]_1$, $[xy]_2$, and $[at]_1 - [ta]_2$, with no $d(a)$ term. If those checks hold, Theorem 3.1 gives the normal form exactly as stated.

What carries the argument

The load-bearing mechanism is the strengthened Composition–Diamond lemma for dialgebras (Theorem 1.6), which works with a deg-lex-center ordering on normal diwords $[u]_m$: here $u$ is an associative word and $m$ is the position of the distinguished center. Under this ordering, a monic set $S$ is a Groebner–Shirshov basis if and only if every composition is trivial modulo $S$, and this is equivalent to the irreducible diwords $\operatorname{Irr}(S)$ forming a $K$-basis of the quotient. The paper's work is to list all intersection compositions of the five families in $S$ and show each reduces to zero; the normal form in Theorem 3.1 is then the complement of the leading diwords of all normal $S$-diwords in this ordering.

What would settle it

Take a dialgebra with a basis element $a$ and a nonzero derivation $d(a)=b$, perhaps with $b$ a new basis element, and recompute the compositions involving the relation $[at]_1-[ta]_2-[b]$. If the reduction produces a leading term not appearing in the set $\operatorname{Irr}(S)$ of Theorem 3.1, for instance any normal diword containing $b$, then the claimed basis and normal form are incomplete.

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Extended reading notes

Core claim

The central claim is Theorem 3.1: the HNN-extension $D^*_d$ presented by $\langle Di, t \mid a \dashv t - t \vdash a = d(a), a \in A\rangle$ has normal form $\operatorname{Irr}(S) = \{[z_m \cdots z_1 x y_1 \cdots y_n]_{m+1} \mid z_j, x, y_i \in X,\ z_{j+1}z_j \neq xx, xy, xt;\ y_i y_{i+1} \neq xx, xy, xt;\ z_1 x \neq xx, xy, xt;\ x y_1 \neq xx, xy, xt,\ \text{for } x > y\}$, where $S = \{[xx]_1, [xx]_2, [xy]_1, [xy]_2, [at]_1 - [ta]_2\}$ and $x,y \in X$, $a \in A$. The paper verifies that every composition of these strong monic polynomials is trivial modulo $S$, so by the Composition–Diamond lemma $S$ is an explicit Groebner–Shirshov basis and $\operatorname{Irr}(S)$ is a $K$-basis of the quotient dialgebra. Corollary 3.1.1 then says the original dialgebra $Di$ embeds into $D^*_d$, because each element of $X$ is already an irreducible normal word.

Load-bearing premise

The load-bearing premise is that the derivation $d$ in the defining relation $a \dashv t - t \vdash a = d(a)$ is identically zero on $A$, because the set $S$ used in the Groebner–Shirshov basis omits the $d(a)$ term and every composition check in the paper is performed on that reduced relation.

Editorial extensions

If this is right

  • Every element of the HNN-extension has a unique expression as a $K$-linear combination of the listed diwords, so the word problem for these quotients is solved.
  • The base dialgebra $Di$ embeds in $D^*_d$, giving the dialgebra version of the classical HNN embedding phenomenon.
  • The explicit restrictions in the normal form—no $xx$, $xy$, or $xt$ blocks in the listed positions—give a direct criterion for when a word is reducible.
  • The same Composition–Diamond-lemma route can produce normal forms for other dialgebra presentations once a candidate set $S$ is supplied.

Reading between the lines

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

  • Inference: because $S$ omits the $d(a)$ term, the normal form as stated applies to the case $d=0$; a nonzero derivation would require enlarging $S$ by the full relation $[at]_1-[ta]_2-[d(a)]$ and recomputing the composition checks.
  • Inference: the intended bridge to HNN-extensions of Leibniz algebras needs derivations that are typically nonzero, so that bridge likely needs the repaired basis rather than the present one.
  • Inference: a concrete test is to repeat the composition checks for an inner derivation $d(a)=a \dashv b - b \vdash a$; if any composition becomes nontrivial modulo the enlarged $S$, the normal form in Theorem 3.1 must gain extra summands.
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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 / 3 minor

Summary. The paper claims to construct an explicit Groebner-Shirshov basis for the HNN-extension of a dialgebra and to determine a normal form for the extension, using the Composition-Diamond lemma of Zhang and Chen. The main result is Theorem 3.1, which states that Irr(S) is a normal form for D*_d, and Corollary 3.1.1, which asserts that the base dialgebra Di embeds into D*_d. The proof is based on a set S defined in Section 3 and a list of composition checks intended to show S is a Groebner-Shirshov basis.

Significance. If the main claim were correct, the paper would provide a useful explicit normal form for HNN-extensions of dialgebras and an embedding theorem, with potential consequences for Leibniz algebras through the dialgebra-Leibniz connection. The paper also advertises a recent Composition-Diamond lemma and attempts an explicit computation. However, the central construction is not supported: the set S does not encode the defining relation of the HNN-extension, and the stated normal form is for a different quotient. The paper does not provide machine-checked proofs or a parameter-free derivation; the central derivation gap is fundamental.

major comments (3)
  1. [Section 3, definition of S] The defining relation of the HNN-extension in (2.1) is a ⊣ t − t ⊢ a = d(a), i.e., [at]_1 − [ta]_2 − d(a) = 0. The set S contains only the polynomial m = [at]_1 − [ta]_2, omitting the term d(a). Unless d(a) = 0 for every a ∈ A, the polynomial m is not a consequence of the defining relation, because d(a) is a length-one polynomial and cannot be generated by the length-two polynomials listed in S. Consequently, every composition check in Section 3 is performed for the relation with d = 0, and the resulting Groebner-Shirshov basis, if valid, applies to a different quotient, not to D*_d.
  2. [Section 3, definition of S] The set S also includes [xx]_1, [xx]_2, [xy]_1, and [xy]_2 for all x, y ∈ X. These polynomials impose x ⊣ y = 0 and x ⊢ y = 0 on the base dialgebra. The presentation (2.1) imposes no such zero products; it only adds the stable-letter relations to an arbitrary dialgebra Di, whose multiplication may be nontrivial. Thus the quotient Di⟨X|S⟩ is not D*_d but a quotient in which the image of Di has trivial multiplication. This alone invalidates Theorem 3.1 as a normal form for D*_d.
  3. [Theorem 3.1] The asserted normal form Irr(S) contains no words involving t: it consists only of elements [z_m ... z_1 x y_1 ... y_n]_{m+1} with z_j, x, y_i ∈ X. An HNN-extension generated by X and t must have basis elements containing t unless the stable letter becomes trivial or is eliminated by the relations. The absence of t from Irr(S) suggests that the relations in S force t to act trivially or to be expressible through X, which is inconsistent with the intended HNN-extension and with the claimed embedding of Di into D*_d. Corollary 3.1.1 is therefore not established by the preceding argument.
minor comments (3)
  1. [Abstract] There are typographical errors: 'digebras' should be 'dialgebras', and 'Groeber-Shirshov' is misspelled.
  2. [References] Reference [3] contains a duplicated author list ('L.A. Bokut, Y. Chen, L.A. Bokut, Y.Q. Chen, C.H. Liu'), and the journal 'Israil J. Math.' in [12] should be 'Israel J. Math.'
  3. [Theorem 3.1] The condition 'for x > y' at the end of the theorem is ambiguous: it is unclear whether x > y is required for all occurrences of x and y in the word or only for the final factor, and the order relations involving z_j and y_i are not fully specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's derivation is self-contained and driven by an external CD lemma; its failures are mathematical errors, not circular reductions.

full rationale

The main derivation chain is: define the HNN-extension by the presentation (2.1), propose a candidate set S, verify all compositions with respect to the CD-lemma from [13], and conclude that Irr(S) is a normal form. Each step is executed inside the paper or imported from an external lemma; no parameter is fitted to a target output and no prediction is a renamed input. The author's coauthorship of [6] and [7] is contextual: [6] supplies the construction of HNN-extensions for dialgebras and [7] supplies the Lie-superalgebra analogue, but Theorem 3.1 does not reduce to either citation. The real defects are non-circular mathematical errors: the defining relation a⊣t − t⊢a = d(a) is represented in S only by [at]1 − [ta]2, dropping d(a); and S contains [xy]1 and [xy]2 for all x,y∈X, which imposes zero products not present in the given dialgebra. Consequently S is not a Groebner-Shirshov basis of the ideal defined by (2.1), so Theorem 3.1 describes a different quotient. That is a correctness gap, not a circularity, because the false step is a mistaken identification of ideals rather than a definitional or fitted reduction of the claimed conclusion to its assumptions.

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

The main unproved input is the Composition-Diamond Lemma from [13]. The paper also assumes the HNN presentation from [6] and the ordering conditions. No free parameters or invented entities appear. The paper implicitly assumes that A consists of basis letters in the composition checks, an ad hoc assumption.

assumptions (4)
  • standard math Composition-Diamond Lemma for dialgebras (Theorem 1.6) from Zhang and Chen [13]
    The equivalence between a Groebner-Shirshov basis and the irreducibles forming a basis is imported without proof, and the paper relies on it for the main conclusion.
  • domain assumption The presentation (2.1) of HNN-extension is the correct object
    Taken from [6], the paper does not justify it, and the subsequent set S does not match it because d(a) is dropped.
  • domain assumption The deg-lex-center ordering on normal diwords is compatible with the dialgebra operations
    Section 1 assumes this ordering, and the composition triviality computations depend on its compatibility with the algebraic operations.
  • ad hoc to paper Every a in A may be treated as a single basis letter x in X
    In Section 3, compositions such as f∧m and h∧m substitute x and y for a, which is only valid if the basis elements are in A; A is a general subalgebra.

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Cite this review

Pith. "Pith review of A Normal form for HNN-extension of Dialgebras." pith.science (2026). https://pith.science/paper/MBTLMOB3

@misc{pith2026190808397,
  author       = {Pith},
  title        = {Pith review of: A Normal form for HNN-extension of Dialgebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBTLMOB3}},
  note         = {Machine review of arXiv:1908.08397}
}
read the original abstract

We consider a new version of Composition-Diamond Lemma for dialgebras in order to obtain an explicit Groebner-Shirshov basis for HNN-extension of dialgebras and determine a normal form for that.

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Works this paper leans on

13 extracted references · 12 canonical work pages

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    journal.png

    G. Zhang, Y. Chen, A New Composition-Diamond Lemma for Di algebras, Algebra Colloquium, 24 (2) , (2017), 232–350. Departamento de Matemtica, UFBA, A venida Adhemar de Barros, 40 .170.110, Sal- vador, BA, Brazil E-mail address : chia.zargeh@ufba.br This figure "journal.png" is ...

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