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The space of Conradian left-preorders

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Pith's one-line read Conradian left-preorder spaces are either finite or uncountable.

desk verdict Genuine generalization of Rivas's theorem with a real but repairable flaw in Proposition 5.9; the main dichotomy and cardinality formulas survive. read the letter →

arxiv 2506.18573 v2 pith:VLHE6FDL submitted 2025-06-23 math.GR

Iván Chércoles-Cuesta This is my paper · ORCID
classification math.GR MSC 06F1520F60
keywords Conradianleft-preordersspaceofrationalseriesconvexsubgroupsleft-orderablegroupsfiniteoruncountabledichotomyno-abelian-jumps
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 defines Conradian left-preorders, a weakening of Conradian left-orders where the strict cone can have a nontrivial kernel, and studies the space CO(G) of all such structures on a group G. Its main result is a dichotomy and a finiteness criterion: either CO(G) is finite, or it is uncountable; finiteness happens exactly when there is a unique finite rational series through the group, starting at the minimal subgroup C0 that supports relative Conradian preorders, with no abelian jumps at any two consecutive steps. In the finite case the count is forced to be 2^(n+1)-2, where n is the length of that series, and the space of relative Conradian preorders on C0 contributes exactly 2^n of those. The paper also characterizes Conradian left-preorders by convex-jump archimedean embeddings, by an inequality with exponent 2, and by a no-crossing dynamical condition on the action on cosets.

What carries the argument

The load-bearing objects are Conradian left-preorders: total, transitive, reflexive, left-invariant relations on G whose strict cone is such that for any two positive elements α and β some power α^n exceeds β. A rational series is a finite normal chain G0⊴...⊴Gn=G with each quotient torsion-free abelian of rank 1; an abelian jump is a two-step segment Gi⊴Gi+1⊴Gi+2 whose outer quotient Gi+2/Gi is abelian. The proof mechanism is the pair of mutually inverse gluing and splitting maps µ and ρ, which decompose a Conradian preorder on G into relative preorders on the quotient steps of a convex chain; the key theorem is that every term of a rational series without abelian jumps is convex in every Conradian left-preorder. That convexity theorem forces the finite case to look like a binary tree of choices, one sign per step, giving 2^n relative preorders and 2^(n+1)-2 global ones.

What would settle it

Enumerate all Conradian left-preorders on the Klein-bottle group K=<a,b | $aba^{{-1}}$=$b^{{-1}}$>. The theorem predicts exactly 6, or $2^{3}$-2, with binary sign choices for the generators; finding any additional Conradian left-preorder, or showing that one of the six positive cones listed in the paper is not actually convex, would refute the counting part of the main theorem.

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Extended reading notes

Core claim

The central claim is that the Conradian left-preorder space is a two-tier object. For any group with at least one Conradian left-preorder, define C0 as the intersection of all subgroups that support such a preorder; then CO(G) is finite if and only if there is a unique finite rational series C0=G0⊴...⊴Gn=G in which each successive quotient is torsion-free abelian of rank 1 and no two consecutive steps compose to an abelian quotient. When this happens, |CO(G)|=2^(n+1)-2 and |CO_C0(G)|=2^n; when it does not, CO(G) is uncountable. The proof builds a bijection between Conradian preorders respecting a convex chain and tuples of relative preorders on successive quotients, and shows that failure of the no-abelian-jumps condition leaves the space without isolated points, forcing uncountability (and, in the countable case, a Cantor space).

Load-bearing premise

The finiteness criterion rests on Proposition 4.5, which asserts that in a rational series without abelian jumps every intermediate subgroup is convex in every Conradian left-preorder; if the lemma supplying a non-commuting pair across adjacent quotients or the archimedean embedding of each convex jump fails, the formula |CO(G)| = 2^(n+1)-2 no longer follows.

Editorial extensions

If this is right

  • If CO(G) is infinite, it cannot be countably infinite: it is uncountable, and for countable G it is a Cantor set when it has no isolated points.
  • Finiteness of CO(G) is equivalent to all its elements being isolated, and also to the existence of a unique maximal-length rational series with no abelian jumps ending at C0.
  • If G admits a Conradian left-order, then CO(G) is finite exactly when there is a rational series from {1} to G with no abelian jumps, and in that case the number of Conradian left-preorders is 2^(n+1)-2 while the number of Conradian left-orders is 2^n.
  • For the groups T_n generated by the relations a_{i+1} a_i a_{i+1}^{-1} = a_i^{-1}, every left-preorder is Conradian and there are exactly 2^(n+1)-2 of them.
  • Conradian left-preorders admit a dynamical characterization: the left action on G/C has no crossing, which allows the Conradian property to be detected through an action rather than through cone inequalities alone.

Reading between the lines

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

  • Because the finiteness criterion is phrased in terms of a unique rational series, finite CO(G) forces a rigid normal filtration; one natural test is whether any group with finite CO(G) must be poly-(torsion-free abelian of rank 1) with no abelian two-step quotients.
  • The dichotomy may extend beyond Conradian preorders: for any reasonable class of left-preorders defined by forbidding a localized dynamical pattern, a failure of finiteness should again produce uncountably many elements, mirroring Cantor-set phenomena in spaces of left-orders.
  • The cardinality formula suggests a direct enumerative algorithm for finitely presented groups: search normal chains, check the no-abelian-jumps condition, then count sign choices per step; verifying this on small examples such as the Klein-bottle group is a cheap computational check.
  • The no-crossing characterization may connect to harmonic actions on the line: crossing-free actions on coset spaces could supply a dynamical proof of local-indicability-type statements for Conradian preorders.
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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

1 major / 5 minor

Summary. The paper defines Conradian left-preorders (total left-invariant preorders satisfying a Conrad-type condition) and studies the space CO(G) of such preorders on a group G. The main results are: a dichotomy theorem (CO(G) is either finite or uncountable, Corollary 5.13); a characterization of finiteness of CO(G) in terms of existence and uniqueness of a maximal-length rational series with no abelian jumps, together with the explicit cardinality formula |CO(G)| = 2^{n+1}-2 (Theorem 5.10); a relative version for CO_C(G) (Theorem 4.13); and characterizations of Conradian left-preorders via convex jumps, the inequality α ≺ βα², and a no-crossing dynamical condition (Theorems 2.10 and 6.2). The proofs follow and extend the strategy of Rivas for Conradian left-orders.

Significance. If the central results hold, the paper is a substantive contribution to the theory of left-preorders and Conradian orderability. The finiteness criterion and the dichotomy finite/uncountable are new for Conradian left-preorders, and the cardinality formula generalizes Rivas's results for Conradian left-orders. The paper also provides useful characterizations (convex jumps, n=2 condition, dynamical no-crossing) that are natural extensions of known results. The exposition is generally clear and the proofs are detailed; however, the current proof contains a false lemma (Proposition 5.9) that is load-bearing for the main theorem, so the manuscript requires revision before the central claims can be accepted.

major comments (1)
  1. [Proposition 5.9 and its use in Theorem 5.10] Proposition 5.9 is false as stated. For G = Z², the series {0} ≤ ⟨e1⟩ ≤ Z² and {0} ≤ ⟨e1+e2⟩ ≤ Z² are both rational, both have maximal length 2, and are distinct, so uniqueness of a maximal-length rational series fails in general. The proof's step that "CO_{G0}(G) is finite by Theorem 4.10" is invalid because Theorem 4.10 requires the rational series to have no abelian jumps, and a maximal-length series need not satisfy this; indeed, in the counterexample the series has an abelian jump and CO_{G0}(Z²) = CO_{0}(Z²) is infinite. Since Proposition 5.9 is invoked in the proof of Theorem 5.10 to obtain the equivalence (b) ⇔ (c), the proof of the main theorem is currently incomplete. The statement can likely be repaired by proving uniqueness only for maximal rational series satisfying the additional hypotheses that appear in condition (b) (no abelian jumps and CO(G0)=∅), using Lemma 5.6 and Proposition 4.12, but as written the paper contains a false result that must be corrected.
minor comments (5)
  1. [Remark 1.17] The displayed implication "αC⪯βC =⇒ αK ˆ⪯βK" is confusing and appears to be vacuous for α,β ∈ K, since then αK = βK; the intended statement likely concerns α,β ∈ L, and the quantifiers should be clarified.
  2. [Lemma 2.6] In the proof of Lemma 2.6, the Conradian condition is applied to α^m with m = 0, where α^m = 1 is not strictly positive; the case m = 0 should be handled separately or the quantifier restricted to m > 0 to make the argument fully rigorous.
  3. [Lemma 5.6] The sentence "the restriction of ⪯ on G0 give us a Conradian left-preorder on G relative to C0" is garbled; it should state that the restriction gives a Conradian left-preorder on G0 relative to C0.
  4. [Theorem 5.10, condition (d)] In the statement of condition (d), the phrase "There exists a finite rational series finite rational series starting on G" contains a duplicated phrase that should be removed.
  5. [Example 5.17] The assertions that every left-preorder on T_n has stabilizer of the form ⟨a1,...,a_k⟩ and that the number of left-preorders is at most 2^{n-k} are stated without proof; since this example is used to identify all left-preorders, a proof or a precise reference to [8] would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the finiteness and cardinality theorems are derived from definitions plus standard external theorems (Hölder, Sikora, Rivas's proof strategy), with no fitted parameter renamed as a prediction and no self-citation chain.

full rationale

The paper's central claims (Theorem 5.10, Corollary 5.13) are proved from the definition of Conradian left-preorder, the bijective decompositions of Propositions 2.13 and 2.14, a compactness argument, and standard external theorems such as Hölder's theorem and Sikora's topology on spaces of orders. The subgroup C0 is defined as the intersection of all subgroups admitting a Conradian left-preorder relative to them; this is an object constructed from the hypothesis, not a parameter fitted to force the cardinality formula. The formula |CO(G)| = 2^{n+1}-2 is derived by summing the counts |CO_{G_i}(G)| = 2^{n-i}, which in turn come from the bijective decomposition of Proposition 2.14 and the fact that each rank-one quotient admits exactly two orders (Example 3.8). No step identifies a fitted quantity with the quantity being predicted. The introduction and Section 4 openly credit Rivas [17] for the proof strategy for the relative case, but the present paper re-proves the needed arguments internally rather than importing a uniqueness theorem as an unverified black box, so the citation is not load-bearing circularity. The skeptical concern about Proposition 5.9 is a correctness issue, not circularity: the uniqueness claim for maximal-length rational series omits the no-abelian-jumps hypothesis, and its proof incorrectly invokes Theorem 4.10 in a case where that theorem does not apply. That is a gap to be repaired in the proof, but it does not make the main result reduce to its own inputs. Apart from standard cited theorems, the derivation chain is self-contained, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

Pure mathematics paper. The only new object is the definition of Conradian left-preorders, which is a mathematical definition rather than a postulated entity. No free parameters or fitted constants appear. The axioms used are standard set theory, topology, and known theorems in ordered groups, all cited.

assumptions (7)
  • standard math Zermelo-Fraenkel set theory with Choice (ZFC)
    All group-theoretic and topological proofs are carried out in standard ZFC.
  • standard math Tychonoff's theorem: product of compact spaces is compact
    Used in Proposition 3.4 and Proposition 5.2 to prove compactness of CO_C(G) and to construct an element in the intersection via finite intersection property.
  • standard math Holder's theorem: every archimedean left-ordered group embeds into (R,+)
    Cited as Theorem 2.9 and used in Theorem 2.10(c implies a) to obtain order-preserving embeddings L/K into R.
  • standard math Sikora's classification: the space of left-orders on a torsion-free abelian group is two points for rank 1 and a Cantor set for rank greater than 1
    Cited as [18] and used in Example 3.8, Proposition 4.9, and Proposition 4.12.
  • standard math A compact Hausdorff totally disconnected space with no isolated points is uncountable
    Used in Corollary 4.16 to show CO_C(G) is uncountable when not finite.
  • standard math Every subgroup of a left-orderable group is left-orderable
    Used implicitly in Example 5.17 to extend left-preorders to left-orders via a lexicographic construction.
  • domain assumption Left-preorders are non-trivial total left-invariant preorders; the trivial preorder is excluded by definition
    Definition 0.1; this convention ensures the kernel C is a proper subgroup.

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Pith. "Pith review of The space of Conradian left-preorders." pith.science (2026). https://pith.science/paper/VLHE6FDL

@misc{pith2026250618573,
  author       = {Pith},
  title        = {Pith review of: The space of Conradian left-preorders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLHE6FDL}},
  note         = {Machine review of arXiv:2506.18573}
}
read the original abstract

We define Conradian left-preorders and the space of Conradian left-preorders. We show that this space is either finite or uncountable. We describe conditions that are equivalent to say that the space of Conradian left-preorders is finite. We provide some characterizations of Conradian left-preorders.

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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. Full citation record

  1. Spaces of left-preorders on free products

    math.GR 2025-07 conditional novelty 6.0 of 10

    Left-preorder spaces of free products have no isolated elements; when the factors are finitely generated, every nonempty such space is a Cantor set.

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Works this paper leans on

18 extracted references · 3 canonical work pages · cited by 1 Pith paper

  1. [1]

    & Rivas, C

    Antol´ ın, Y. & Rivas, C. The space of relative orders and a generalization of Morris indicability theorem. J. Topol. Anal.. 13, 75-85 (2021), https://doi.org/10.1142/S1793525320500053

    doi:10.1142/s1793525320500053 2021

  2. [2]

    &ˇSuni´ c, Z

    Antol´ ın, Y., Dicks, W. &ˇSuni´ c, Z. Left relatively convex subgroups.Topological Methods In Group Theory. 451 pp. 1-18 (2018)

    work page 2018

  3. [3]

    & Triestino, M

    Brum, J., Bon, N., Rivas, C. & Triestino, M. Locally moving groups and laminar actions on the line. (2024), https://arxiv.org/abs/2104.14678

    arXiv 2024

  4. [4]

    & Rolfsen, D

    Clay, A. & Rolfsen, D. Ordered groups and topology. (American Mathematical Society, Providence, RI,2016), https://doi.org/10.1090/gsm/176

    doi:10.1090/gsm/176 2016

  5. [5]

    Right-ordered groups

    Conrad, P. Right-ordered groups. Michigan Math. J. . 6 pp. 267-275 (1959), http://projecteuclid.org/euclid.mmj/1028998233

    arXiv 1959

  6. [6]

    & Rond, G

    Decaup, J. & Rond, G. Preordered groups and valued fields. (2019), https://arxiv.org/abs/1912.03928 34 IV ´AN CH ´ERCOLES CUESTA

    arXiv 2019

  7. [7]

    & Hurtado, S

    Deroin, B. & Hurtado, S. Non left-orderability of lattices in higher rank semi-simple Lie groups. (2020), https://arxiv.org/abs/2008.10687

    arXiv 2020

  8. [8]

    & Rivas, C

    Deroin, B., Navas, A. & Rivas, C. Groups, Orders, and Dynamics. (2016), https://arxiv.org/abs/1408.5805

    arXiv 2016

Show all 18 references
  1. [9]

    & Ishida, M

    Ewald, G. & Ishida, M. Completion of real fans and Zariski-Riemann spaces. Tohoku Math. J. (2) . 58, 189-218 (2006), http://projecteuclid.org/euclid.tmj/1156256400

    2006

  2. [10]

    & Teissier, B

    Gonz´ alez P´ erez, P. & Teissier, B. Toric geometry and the Semple-Nash modification.Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM . 108, 1-48 (2014), https://doi.org/10.1007/s13398-012-0096-0

    2014 doi

  3. [11]

    & Young, G

    Hocking, J. & Young, G. Topology. (Dover Publications, Inc., New York,1988)

    1988

  4. [12]

    Grupos ordenables: estructura algebraica y din´ amica

    Jim´ enez, L. Grupos ordenables: estructura algebraica y din´ amica. Master thesis, Univ. de Chile (2008)

    2008

  5. [13]

    Amenable groups that act on the line

    Morris, D. Amenable groups that act on the line. Algebr. Geom. Topol. . 6 pp. 2509-2518 (2006), https://doi.org/10.2140/agt.2006.6.2509

    2006 doi

  6. [14]

    Topology

    Munkres, J. Topology. (Prentice Hall, Inc., Upper Saddle River, NJ,2000)

    2000

  7. [15]

    On the dynamics of (left) orderable groups

    Navas, A. On the dynamics of (left) orderable groups. Ann. Inst. Fourier (Grenoble) . 60, 1685-1740 (2010), https://doi.org/10.5802/aif.2570

    2010 doi

  8. [16]

    & Rivas, C

    Navas, A. & Rivas, C. A new characterization of Conrad’s property for group orderings, with applications. Algebr. Geom. Topol.. 9, 2079-2100 (2009), https://doi.org/10.2140/agt.2009.9.2079, With an appendix by Adam Clay

    2009 doi

  9. [17]

    On spaces of Conradian group orderings

    Rivas, C. On spaces of Conradian group orderings. J. Group Theory . 13, 337-353 (2010), https://doi.org/10.1515/JGT.2009.053

    2010 doi

  10. [18]

    Topology on the spaces of orderings of groups

    Sikora, A. Topology on the spaces of orderings of groups. Bull. London Math. Soc. . 36, 519-526 (2004), https://doi.org/10.1112/S0024609303003060 Instituto de Ciencias Matem´aticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Investigaciones Cient´ıficas, C/ Nicol´as Cabrera, 13–1...

    2004 doi

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