{"id":"6321b300-0fa2-4e9e-859f-d0a8e8cde052","arxiv_id":"1908.08397","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims an explicit Groebner-Shirshov basis and normal form for HNN-extensions of dialgebras, but omits the derivation term from the defining relations and the new generator from the stated normal form.","lead":"This paper claims to give an explicit Groebner-Shirshov basis and a normal form for HNN-extensions of dialgebras, a class of non-associative algebras related to Leibniz algebras. It is a candidate reference for rewriting methods in combinatorial algebra, but the main proof omits a key term from the defining relations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proposed set S does not generate the defining ideal of D*_d: it drops d(a) from the HNN relation and sets every product of base elements to zero, so Theorem 3.1's normal form is for a different quotient.","rationale":"The reader's identification of the omitted d(a) is correct and is already fatal. I would put the same fact in a slightly broader frame: the condition that S must generate the ideal of the HNN presentation fails in two independent places. Even setting d=0, the polynomials [xy]_1 and [xy]_2 are not consequences of (2.1) and force the base dialgebra to have zero product; this alone makes the claimed normal form incompatible with an arbitrary dialgebra Di. The concrete test isolates the d-omission in the simplest case: a one-generator free dialgebra with d(x)=x gives a presentation whose defining relation is not in Id(S). The paper's composition table therefore establishes triviality for a different ideal, and Theorem 3.1 cannot be a normal form for D*_d. I do not see a salvage short of replacing S with the actual multiplication table of Di and with m−d(a), and redoing all compositions; the current text gives no such set and no proof.","tokens_in":7007,"tokens_out":14296,"duration_ms":156473,"concrete_test":"Take Di to be the free dialgebra on one generator x, A=Di, and the derivation d determined by d(x)=x (extended by the derivation rules). The defining relation (2.1) is [xt]_1 − [tx]_2 − x = 0. In the quotient by S, S contains [xt]_1 − [tx]_2 and the length-two words [xx]_1, [xx]_2 = 0, so all leading monomials in S have length at least two. The word [x]_1 is therefore nonzero in Di⟨X'|S⟩ by the deg-lex-center ordering (and by Theorem 1.6 if S were a GS basis). Hence [xt]_1 − [tx]_2 − x equals −x, which is nonzero in this quotient. The defining relation of D*_d is not satisfied, so the quotient by S is not D*_d.","verdict_should_be":"UNCHANGED","load_bearing_attack":"For Theorem 3.1 to be a normal form for D*_d, S must be a Groebner-Shirshov basis of the ideal generated by the presentation (2.1) in the free dialgebra on X∪{t}. The displayed S is not contained in this ideal. The defining relation is a⊣t − t⊢a = d(a), i.e. [at]_1 − [ta]_2 − d(a) = 0, while S contains only m = [at]_1 − [ta]_2. Unless d(a)=0 for every a∈A, m is not a consequence of the defining relation: length-one elements such as d(a) are not in the ideal generated by the length-two polynomials listed. Independently, S also contains [xy]_1 and [xy]_2 for every x,y∈X, imposing x⊣y = x⊢y = 0 on the base dialgebra. The presentation (2.1) imposes no such zero products; it only adds stable-letter relations to a given (possibly nonzero) dialgebra multiplication. Thus every composition check in Section 3 is performed against the wrong ideal. The resulting Irr(S) may describe a quotient of the free dialgebra, but it is not the HNN-extension D*_d, and Corollary 3.1.1 does not follow.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7301,"tokens_out":3260,"duration_ms":34104,"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":[{"comment":"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.","section":"Section 3, definition of S"},{"comment":"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.","section":"Section 3, definition of S"},{"comment":"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.","section":"Theorem 3.1"}],"minor_comments":[{"comment":"There are typographical errors: 'digebras' should be 'dialgebras', and 'Groeber-Shirshov' is misspelled.","section":"Abstract"},{"comment":"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.'","section":"References"},{"comment":"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.","section":"Theorem 3.1"}],"recommendation":"reject","confidential_remarks":"The manuscript appears to be an early-stage research note. The main error is not a local gap but a mismatch between the proposed Groebner-Shirshov basis S and the defining ideal of D*_d; this cannot be repaired by a modest revision within the current scope. The paper also does not compare with the existing HNN-extension construction in [6], despite sharing an author."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Thanks for sharing the report. I read the paper alongside the stress-test note, and the note lands. The set S in Section 3 is not a subset of the ideal of the presentation (2.1), so the central claim does not survive.\n\nWhat is genuinely there: the paper states the HNN-extension construction for dialgebras, with the derivation d in the stable-letter relation, and attempts the standard Groebner-Shirshov approach. The composition checks are written out explicitly, so a reader can follow what the author computed. The recall of the Zhang-Chen Composition-Diamond lemma is competent. That is about where the credit stops.\n\nThe problem is structural. (2.1) reads a⊣t − t⊢a = d(a). The set S contains [at]1 − [ta]2 but drops the d(a) term. Unless d is identically zero, that length-one element is not in the ideal generated by the length-two polynomials listed, so the basis is for a different presentation. On top of that, S includes [xy]1 and [xy]2 for all x,y in X, which forces x⊣y = x⊢y = 0. The HNN presentation imposes no such zero products on Di; it only adds stable-letter relations to the existing dialgebra multiplication. Every composition check in Section 3 is performed against this wrong ideal. Theorem 3.1 then gives a normal form with no word containing t, which cannot be a basis of an extension that contains t. Corollary 3.1.1 does not follow. This is not a minor omission; the quotient Irr(S) is not the HNN-extension D*_d.\n\nThe paper is short, has no code or machine-checked proof, and leans on the author's own [6] for the construction. That alone would not be a problem, but here the derivation gap is the load-bearing failure.\n\nWho gets value from this? Someone studying the Zhang-Chen version of the CD lemma might find the composition list a useful worked exercise. As a paper claiming a normal form for HNN-extensions of dialgebras, it does not deserve referee time. The mismatch between (2.1) and S would be flagged by any specialist on the first read. I would desk reject and invite a correction or an explicit hypothesis d=0 in a resubmission.\n\nRecommendation: don't send this forward for peer review in its current form.","headline":"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.","tokens_in":7812,"tokens_out":3071,"would_cite":false,"duration_ms":29798,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17A36","17A50","17A99","13P10","16S15"],"pacs":[],"model":"deepseek-v4-flash","headline":"An explicit Groebner–Shirshov basis and normal form for HNN-extensions of dialgebras","keywords":["normal form","dialgebra","diassociative algebra","Groebner-Shirshov basis","HNN-extension","Composition-Diamond lemma","derivation"],"falsifier":"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.","tokens_in":6797,"feed_emoji":"🧮","tokens_out":8552,"duration_ms":77301,"temperature":0.7,"pith_summary":"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.","feed_headline":"Normal form derived for HNN-extensions of dialgebras","feed_subtitle":"A Groebner–Shirshov basis gives a linear basis and embedding—if the derivation is zero.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the strengthened Composition-Diamond lemma with the monomial-center ordering used to equate a Groebner-Shirshov basis with a normal-form basis.","marker":"[13]"},{"why":"provides the earlier Composition-Diamond lemma for dialgebras and the normal-diword basis of the free dialgebra that the paper starts from.","marker":"[3]"},{"why":"introduces the construction and presentation of HNN-extensions of dialgebras and of Leibniz algebras that this paper extends.","marker":"[6]"},{"why":"gives the original group-theoretic HNN-extension theorem whose embedding conclusion is the analogue being proved here.","marker":"[5]"},{"why":"defines dialgebras and their two associative products, setting the algebraic framework for all normal-form computations.","marker":"[9]"}],"fun_headline_variants":["Explicit normal form for HNN dialgebra extensions","Groebner-Shirshov basis yields dialgebra HNN normal form","HNN dialgebra extensions: explicit normal form","Normal form solves HNN dialgebra embedding"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit normal form for HNN dialgebra extensions","Groebner-Shirshov basis yields dialgebra HNN normal form","HNN dialgebra extensions: explicit normal form","Normal form solves HNN dialgebra embedding"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000687,"raw_usage":{"total_tokens":3076,"prompt_tokens":868,"completion_tokens":2208,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":2143}},"tokens_in":484,"tokens_out":2208,"duration_ms":17177,"temperature":1.0,"reasoning_tokens":2143,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:32:05.444193+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"A Normal form for HNN-extension of Dialgebras","cited_arxiv_id":"1908.08397","evidence_quote":"supplies the strengthened Composition-Diamond lemma with the monomial-center ordering used to equate a Groebner-Shirshov basis with a normal-form basis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the earlier Composition-Diamond lemma for dialgebras and the normal-diword basis of the free dialgebra that the paper starts from."},{"cited_title":"Ladra, M","cited_arxiv_id":null,"evidence_quote":"introduces the construction and presentation of HNN-extensions of dialgebras and of Leibniz algebras that this paper extends."},{"cited_title":"Higman, B.H","cited_arxiv_id":null,"evidence_quote":"gives the original group-theoretic HNN-extension theorem whose embedding conclusion is the analogue being proved here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines dialgebras and their two associative products, setting the algebraic framework for all normal-form computations."}],"review_version":1}