REVIEW 6 major objections 5 minor 18 references
Strongly topologically orderable gyrogroups with a suitable set
T0 review · 6 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read Every locally compact or not totally disconnected strongly topologically orderable gyrogroup is metrizable and contains a suitable set; dense subgyrogroups inherit suitable sets.
desk verdict Solid metrizability reductions for strongly topologically orderable gyrogroups, but the advertised suitable-set existence leans on an unpublished self-cited theorem. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing device is the L-subgyrogroup: a subgyrogroup whose gyration automorphisms keep it invariant, so that its left cosets partition the gyrogroup. In a strongly topologically orderable gyrogroup, the total order is converted into a totally ordered local base at the identity consisting of clopen L-subgyrogroups, and this base does the heavy lifting: it yields hereditary paracompactness, forces first-countability in the not-totally-disconnected case, and in the locally compact totally disconnected case produces a compact clopen L-subgyrogroup that is either discrete or homeomorphic to the Cantor set, giving metrizability.
What would settle it
Exhibit a metrizable strongly topological gyrogroup that has no suitable set; because the paper's corollaries convert the relevant orderable gyrogroups into metrizable ones and then stop, any such example would falsify the suitable-set conclusions and [18, Theorem 1]. A natural place to search is among countable, non-discrete metrizable gyrogroups, where the candidate discrete generating subsets can be listed directly.
Extended reading notes
Core claim
On its own terms, the paper establishes a structural dichotomy: every strongly topologically orderable gyrogroup is either metrizable or carries a totally ordered local base at the identity made of clopen L-subgyrogroups invariant under all gyrations, and in either case it is hereditarily paracompact. From this structure the paper proves that a not-totally-disconnected strongly topologically orderable gyrogroup contains an open first-countable L-subgyrogroup and is metrizable, and that every locally compact strongly topologically orderable gyrogroup is metrizable, with the totally disconnected compact case splitting as either discrete or homeomorphic to the Cantor set. The paper then concludes, using [18, Theorem 1], that both classes admit suitable sets; it also shows that dense subgyrogroups inherit closed suitable sets.
Load-bearing premise
The suitable-set conclusions rest on the cited theorem [18] that every metrizable strongly topological gyrogroup has a suitable set; the paper proves metrizability but not that theorem, so the main corollaries would fail if that external theorem fails.
Editorial extensions
If this is right
- Every not-totally-disconnected strongly topologically orderable gyrogroup contains a suitable set, generalizing the known group case.
- Every locally compact strongly topologically orderable gyrogroup is metrizable and therefore contains a suitable set.
- If the identity element of a strongly topologically orderable gyrogroup is a $G_\delta$-set, then the gyrogroup is metrizable and has a suitable set.
- If a strongly topologically orderable gyrogroup has a closed suitable set, then every dense subgyrogroup of it also has a closed suitable set.
- Every strongly topologically orderable gyrogroup is hereditarily paracompact.
Reading between the lines
- Going beyond the paper: because every topological group is a gyrogroup with trivial gyrations, the same proof supplies suitable sets for all locally compact and all not-totally-disconnected topologically orderable groups, matching the classical result the paper generalizes.
- Going beyond the paper: if the cited theorem that every metrizable strongly topological gyrogroup has a suitable set is fully proved, the paper's route would show that every metrizable strongly topologically orderable gyrogroup has a suitable set, leaving orderability itself as the only hypothesis that matters.
- Going beyond the paper: the totally ordered clopen L-subgyrogroup base constructed here is a concrete structure that might yield suitable sets directly in non-metrizable cases, without passing through metrizability and the external final theorem.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies suitable sets in strongly topologically orderable gyrogroups. It proves a dichotomy: such a gyrogroup is either metrizable or has a totally ordered local base of clopen L-subgyrogroups (Theorem 3.4), gives characterizations of topological orderability for non-metrizable strongly topological gyrogroups (Theorem 3.5), proves hereditary paracompactness (Theorem 3.6), and establishes metrizability of not totally disconnected (Theorem 4.4) and locally compact (Theorem 4.10) strongly topologically orderable gyrogroups. It then concludes these classes have suitable sets using [18, Theorem 1], and proves a hereditary result for dense subgyrogroups (Theorem 5.4).
Significance. If the gaps identified below are repaired, the paper would establish a natural gyrogroup analogue of the classical result that each not totally disconnected topologically orderable group is metrizable and hence has a suitable set, and would extend suitable-set existence to locally compact strongly topologically orderable gyrogroups. The structural dichotomy in Theorem 3.4 and the hereditary paracompactness result in Theorem 3.6 are potentially useful contributions. However, the advertised existence theorems depend at the last step on an unpublished accepted paper by two of the present authors, and the proof of the key orderability characterization in Theorem 3.5 is delegated to the group case without verification for non-associative gyrogroups. The metrizability reductions are plausible, but the paper is not self-contained as it stands.
major comments (6)
- [Theorem 3.5, implication (3)⇒(1)] The construction of the compatible linear order is delegated to [13, Theorem 6] with the phrase 'a complete similar method', but [13] treats topological groups. The non-associative gyrogroup operation affects both the transitivity of the lexicographic order and the verification that the order topology agrees with the given topology; no argument is given that the method transfers. Since Theorem 3.5 is used in Theorem 3.6, Theorem 3.7, and Lemma 5.3, this is a load-bearing gap.
- [Theorem 3.5, implication (2)⇒(3)] The proof states '(2) ⇒ (3) by Theorem 3.4 and Lemma 3.3 respectively', but Theorem 3.4 assumes G is topologically orderable, which is exactly what (2) is supposed to imply in the equivalence being proved. Thus the cited theorem cannot be used in this direction without circularity. A direct proof that a totally ordered local base plus strong gyrogroup invariance yields a totally ordered local base of clopen L-subgyrogroups is missing.
- [Corollary 4.5, Theorem 4.6, Theorem 4.10] Each of these results concludes the existence of a suitable set from metrizability by invoking [18, Theorem 1], an accepted paper by two of the present authors that has no public version and whose statement is only partially quoted. The advertised existence results in the abstract therefore depend on a result that is neither proved nor available for verification. The authors should include the full statement and proof of [18, Theorem 1] or provide a publicly accessible preprint.
- [Lemma 3.3] The construction of the totally ordered invariant local base V is not rigorous. The conditions 'Uασ ⊆ Wαβ ⊆ Uαβ' mix ordinals and sets (with σ=β+1), and the argument does not show that the selected family is a base, is totally ordered by inclusion, or inherits the invariance property. Since Lemma 3.3 underpins Theorem 3.4, this proof needs to be rewritten in detail.
- [Theorem 4.10] In the totally disconnected case the proof infers from Lemma 4.9 that G is metrizable. Lemma 4.9 only provides a clopen L-subgyrogroup H homeomorphic to the Cantor set. The missing steps are that the left cosets of H form an open cover of G by metrizable subspaces, so G is locally metrizable, and by Theorem 3.6 it is paracompact; metrizability then follows from the Smirnov metrization theorem. As written, the inference is unsupported.
- [Lemma 5.3] The statement assumes D⊆L, yet the proof contains the case 'If x∉L' for x∈D, which is impossible under the assumption, and condition (iii) 'D ⊆ F in G' is unclear. The intended statement appears to be about transferring a discrete set D in G to a discrete set F inside a dense subgyrogroup L, but the lemma as stated is internally inconsistent. This affects the validity of Theorem 5.4.
minor comments (5)
- [Throughout] There are numerous typos: 'oederable' in Question 1.1; 'exisits' in Section 2; 'subgyrgroup' in Lemmas 4.8 and 4.9; 'Vedenissov' for Vedenissoff; 'Narural Science' in reference [18]; 'stronglly' in the Section 5 heading.
- [Section 5 introduction] The introduction claims that a strongly topologically orderable gyrogroup has a suitable set if and only if each dense subgyrogroup has one, but Theorem 5.4 proves only the forward direction; the converse is not shown and is not trivial, so either prove it or adjust the claim.
- [Theorem 5.4 proof] The line 'D ⊂ H = G' should be 'D ⊆ H' since H is a dense subgyrogroup and need not equal G.
- [Lemma 3.1] The terms 'cofinality from above' and 'cofinality from below' are used without definition; add a definition or a precise reference to [13]. Also, 'lim dτ = 1' should be phrased as 'dτ → 1' with quantifiers over neighborhoods.
- [Lemma 4.8] The phrase 'zero-dimension' should be 'zero-dimensional'.
Circularity Check
Suitable-set existence for the headline classes is imported from an unavailable self-citation [18, Theorem 1]; the metrizability reductions are independent.
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self citation load bearing
[Section 4, Corollary 4.5, Theorem 4.6, and Theorem 4.10]
"By Theorem 4.4 and Lemma 4.9, G is metrizable. Since each metrizable strongly topological gyrogroup has a suitable set [18, Theorem 1], we conclude G has suitable set."
The abstract advertises exactly the existence of suitable sets for locally compact or not totally disconnected strongly topologically orderable gyrogroups. In the proofs, the only bridge from the newly proved metrizability to the existence of a suitable set is an invocation of [18, Theorem 1], an accepted paper by two of the present authors for which no proof or public text is given here. The paper does not derive the statement 'each metrizable strongly topological gyrogroup has a suitable set'; it imports it verbatim from a self-citation. Thus the headline suitable-set claims are not self-contained: they are as strong as an unavailable result by the same authors.
full rationale
The paper's internal contributions are mostly self-contained: Section 3 derives metrizability, hereditary paracompactness, and local-base characterizations from definitions and standard external citations, and Section 5 proves the dense-subgyrogroup transfer using the paper's own Lemma 5.3. However, every advertised suitable-set existence conclusion for the two headline classes ends by invoking [18, Theorem 1] rather than proving it. Since [18] is by two of the present authors, is accepted but not publicly available, and supplies the exact final step from metrizability to a suitable set, the central suitable-set claim is not independent of the authors' own prior result. This is a missing-support dependency rather than a definitional or fitted-input circularity; the metrizability reductions genuinely carry new content. On the scale, the appropriate finding is partial self-citation load-bearing, not full circularity.
Assumptions & free parameters
assumptions (6)
- domain assumption Every first-countable strongly topological gyrogroup is metrizable.
- domain assumption Every metrizable strongly topological gyrogroup has a suitable set.
- domain assumption Connected subsets of topologically orderable spaces contain non-empty open intervals, and an identity component arising here is locally compact.
- domain assumption Compact connected homogeneous spaces with at least two points are not orderable.
- ad hoc to paper The construction in [13, Theorem 6] of a compatible linear order on a topologically orderable group transfers unchanged to gyrogroups.
- standard math Open L-subgyrogroups have finite index in compact strongly topological gyrogroups, and subgyrogroup properties from [15] and [10] hold.
Cite this review
Pith. "Pith review of Strongly topologically orderable gyrogroups with a suitable set." pith.science (2026). https://pith.science/paper/NDOBMWR5
@misc{pith2026250710909,
author = {Pith},
title = {Pith review of: Strongly topologically orderable gyrogroups with a suitable set},
year = {2026},
howpublished = {\url{https://pith.science/paper/NDOBMWR5}},
note = {Machine review of arXiv:2507.10909}
}
abstract
A discrete subset $S$ of a topologically gyrogroup $G$ is called a {\it suitable set} for $G$ if $S\cup \{1\}$ is closed and the subgyrogroup generated by $S$ is dense in $G$, where $1$ is the identity element of $G$. In this paper, we mainly study the existence of suitable set of strongly topologically orderable gyrogroups, which extends some result in some papers in the literature. In particular, the existences of suitable set of each locally compact or not totally disconnected strongly topologically orderable gyrogroup are affirmative.
Reference graph
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Reviewed August 6, 2026 · model on record in the stance chip above.
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