On a lattice of relational spaces (reducts) for the order of integers

A.L. Semenov; S. F. Soprunov

REVIEW 5 major objections 7 minor 2 cited by

On a lattice of relational spaces (reducts) for the order of integers

T0 review · 5 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that the interval between 1-codirection and the full order relation in the definability lattice of the integers is exactly the finite set of spaces in the Diagram, with strict inclusions along the edges and no other…

desk verdict A plausible classification of the [A11,<] interval for <Z,<> that needs a proper proof of Lemma 4 before I'd trust the exhaustiveness. read the letter →

arxiv 2411.18181 v1 pith:FF5MKRP2 submitted 2024-11-27 math.LO

A.L. Semenov , S. F. Soprunov This is my paper · ORCID

classification math.LO MSC 03C0706A0520B27

keywords definabilitylatticereductsorderofintegers1-codirectionbetweenrelationcycleseparationclosedpermutationgroups

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

The paper studies the definability lattice of the ordered set of integers $\langle \mathbb{Z}, <\rangle$: which relations on the integers can be defined from the order relation, and which of those relations define one another. The main theorem describes a contiguous slice of that lattice, the interval lying between the relation of 1-codirection (two neighboring segments pointing the same way) and the full order relation. The theorem states that the Diagram in the paper lists exactly the definability spaces in that interval, that each edge of the Diagram is a strict inclusion, and that every other inclusion is already forced by transitivity. If this is correct, then any relation definable from the integer order that is at least as expressive as 1-codirection and strictly less expressive than the order itself must coincide with one of the spaces shown. The full lattice for $\langle \mathbb{Z}, <\rangle$ remains open, but this central interval is now completely known.

What carries the argument

The argument is carried by an anti-isomorphism (Proposition 1) between the definability lattice of a structure with an upward complete extension and the lattice of closed supergroups of its automorphism group. For $\langle \mathbb{Z}, <\rangle$ the upward complete extension is $\mathbb{Q} \times \mathbb{Z}$, thought of as copies of $\mathbb{Z}$ (vertical fibers) indexed by $\mathbb{Q}$. The work then classifies closed groups $\Gamma^*$ sitting between the shift group and the automorphism group $\Gamma(A1_1)$ of the 1-codirection relation: each such group is determined by its image under the 'initiation' map to $\operatorname{Sym}(\mathbb{Q})$ together with which elements are initiated by positive versus negative permutations. Proposition 2 limits the possible images to five known groups from the rational order, and Lemma 3 (every definable relation has a finite boundary) together with Lemma 4 (certain permutations acting on four rationals force $\Gamma(+1)$ or $\Gamma(A1_1)$) make each case a finite check. This machinery converts a question about which relations are definable into a question about which closed permutation groups are possible.

What would settle it

Exhibit a closed subgroup of $\Gamma(A1_1)$ that contains the shift group and is not one of the groups corresponding to the Diagram's vertices; its existence would force an additional definability space in $[A1_1, <]$, contradicting the main theorem. A concrete first search would be to compute the automorphism group of the $n$-codirection relation $A1_n$ for $n>1$ and compare it with the Diagram groups; a mismatch would expose a missing space.

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Extended reading notes

Core claim

The main theorem is the completeness statement that the Diagram (Fig. 2) shows all definability spaces greater than $A1_1$ and smaller than $<$, where $A1_1$ is the 1-codirection relation. The authors prove that the interval $[A1_1, <]$ in the definability lattice of $\langle \mathbb{Z}, <\rangle$ is exactly the finite sublattice shown in the Diagram, generated by the relations between, cycle, separation, neighbor, 1-codirection, equality, and order, with strict inclusions along the edges and all remaining inclusions contained in the transitive closure of the edges. The proof works through the upward complete extension $\mathbb{Q} \times \mathbb{Z}$: by Proposition 1 the definability lattice is anti-isomorphic to the lattice of closed supergroups of the shift group, so the problem becomes a classification of the closed permutation groups lying between the shift group and the automorphism group of $A1_1$. Under the 'initiation' map, each such group projects to one of the five closed supergroups of the rational shift group given by Proposition 2, and the five resulting cases are analyzed using the boundary lemma for definable relations. The conclusion is a finite, exact picture of the interval.

Load-bearing premise

The proof depends on two prior results being exactly right: that every definability relation on the integer order is mirrored by a symmetry group of a larger ordered structure, and that the order of the rational numbers has only the five known symmetry groups above its shift maps; if either background classification is incomplete, the five-case analysis could miss a definability space.

Editorial extensions

If this is right

If the main theorem is correct, the interval $[A1_1, <]$ in the definability lattice of $\langle \mathbb{Z}, <\rangle$ is completely known: every definability space in that interval coincides with one of the Diagram's vertices.
Because the inclusions are exactly the transitive closure of the Diagram's edges, the Hasse diagram functions as a complete lookup table for the interval, with no hidden dependencies between vertices.
The five-case group-theoretic analysis establishes that the rational-order lattice and the successor lattice of $\mathbb{Z}$ together control this part of the integer-order lattice, so the method transfers to other intervals with the same two background classifications.
Any future full description of all reducts of the integer order must place the Diagram as a finite block within the larger lattice, which substantially narrows the open problems stated in the paper.
The theorem confirms that the three classical relations between, cycle, and separation, together with neighbor and 1-codirection, generate every definability space in this interval.

Reading between the lines

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

The paper leaves it implicit that the same group-theoretic machinery should classify the interval above the $n$-codirection relation $A1_n$ for each fixed $n$; the boundary lemma and the five-image split do not depend on $n$, so a parallel diagram for each $n$ is a plausible extension.
One testable consequence of the method, not stated in the paper, is that every definability space in the interval $[A1_1, <]$ is finitely determined: the boundary lemma suggests that membership of a relation in the interval can be decided from a finite amount of data about the relation.
If the main theorem is right, the open region below $A1_1$ contains the genuinely intricate part of the integer-order reduct lattice; the authors' own open problems point to the neighborhood relation as the next boundary, and one could try to classify the interval between neighbor and order in the same style.
A computation of the automorphism group of $A1_n$ for $n>1$ would be a direct, minimal test of the completeness claim: if such a group lies strictly between $\Gamma(A1_1)$ and $\Gamma(<)$ and is not one of the Diagram groups, the theorem would need revision.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper's main result is that the interval between 1-codirection and the order relation in the definability lattice for is exactly the diagram in Fig. 2. That is a real and useful step: it connects the known lattices for and and pins down the behavior between two previously understood layers. The automorphism-group method is the same one the authors used for , so the strategy is established rather than novel, but that's fine; the target is the classification, not the machinery.

What I like: the reduction to five initiated subgroups of Sym(Q) via Proposition 2 is clean, and the distinction between positive and negative permutations does the heavy lifting. If the main theorem holds, it fills an open interval and gives a concrete diagram of relations (between, cycle, separation, neighbor, 1-codirection) that people in the area will want to have.

The soft spot is Lemma 4, and it's not minor. The proof claims that from a single positive permutation reversing four verticals you can generate every finite block permutation. The displayed construction for n=3 gives only four of the six permutations; the text just says 'it is clear' that repeated applications give all of them. That is true in the small case, and the underlying operations probably do generate the symmetric group, but the paper doesn't show it. Since Lemma 4 is what rules out mixed-orientation groups and forces the five cases to be exhaustive, this gap sits directly under the main theorem. The later 'maximality' checks for Between, Cycle, and Order are also one-paragraph assertions, and the Order case is essentially a single sentence. These are fixable, but they need to be written out.

I also want to flag the reliance on the authors' own earlier results [1], [3], [4] for the Svenonius correspondence and the Q and Z,+1 lattices. That's normal for this program, and the cited results appear to be established, so I don't hold it against them. But a referee should verify that the anti-isomorphism applies to Q x Z and that Proposition 2 is exactly the five-subgroup list.

Bottom line: the result is plausible and worth a serious referee. I would not desk-reject it. But I'd send it back for a substantial revision of the proof of Lemma 4 and the case checks before accepting. As it stands, I wouldn't cite it as a proven classification, only as a conjecture with a promising proof sketch.

Referee Report

5 major / 7 minor

Summary. The paper studies the definability (reducts) lattice of the ordered set of integers \langle\mathbb{Z},<\rangle. It introduces the relations of 1-codirection (A11), neighborhood, between (B), cycle (C), separation (S), and the order relation itself, and claims that the interval [A11,<] in the definability lattice is exactly the sublattice shown in Figure 2, with strict inclusions along the edges and all other inclusions given by transitive closure. The proof strategy is to use a Svenonius-type correspondence (Proposition 1, from the authors' prior work [1]) to translate the problem into a classification of closed subgroups of the automorphism group of A11 that contain the shift group, then to split into five cases according to the induced subgroup of Sym(\mathbb{Q}) (Proposition 2 from [3]), and to analyze each case to show that the only possible closed groups are those corresponding to vertices of the diagram.

Significance. If the main theorem is correct, it constitutes a substantial step toward a full understanding of the definability lattice of \langle\mathbb{Z},<\rangle, a structure of central interest in the area of reducts and oligomorphic groups. The paper leverages a powerful and elegant correspondence between definability spaces and closed automorphism groups, and it builds naturally on the authors' previous work. The claimed classification of the interval [A11,<] is falsifiable and concrete, and the paper correctly identifies the five possible initiated subgroups of Sym(\mathbb{Q}), which is a well-founded starting point. However, the proof as written is far too compressed: the key technical lemma (Lemma 4) is not rigorously established, and the maximality arguments in several of the five cases are only sketched. Because of these gaps, the paper's central claim is not yet substantiated, though the overall approach appears promising and the gaps seem addressable in a thorough revision.

major comments (5)

[Section 4, Lemma 4] The proof of Lemma 4 is not complete. The central assertion that "by applying such vector transformations for different k,l,m we can obtain every predetermined order of blocks" is unproved. For a vector with n=3 blocks, the displayed construction (*) yields only four of the six possible permutations in a single step, and the text does not show that compositions of these transformations generate the full symmetric group on the blocks. Since Lemma 4 is the engine that rules out mixed-orientation groups and forces each initiated subgroup to have the orientation pattern claimed in the diagram, this gap is load-bearing. The authors must either provide a proof that the transformations (*) generate the symmetric group, or replace the argument with a different one.
[Section 4, Lemma 4 (continued)] Even if all finite block permutations were generated, the inference from block-permutation invariance to the conclusion Γ* ⊃ Γ(+1) (or Γ* ⊃ Γ(A11) in case (ii)) is not justified. The proof states "Thus, any relation preserved by Γ* is preserved by any finite permutation of Γ(+1)" and then uses closure to conclude Γ* contains Γ(+1), but it does not explain why the finite block permutations form a dense subgroup of Γ(+1), nor does it define the "family of relations defined by +1" that the closure argument refers to. A careful density or definability argument is needed here.
[Section 4, "Order" case] The proof of maximality for the group corresponding to the initiated subgroup Γ(Q,<) is a single sentence and is not a proof. It asserts that any proper supergroup is "considered above" or has a negative element leading to Sym(Q) by Lemma 4(ii), but it does not explain why a supergroup that initiates exactly Γ(Q,<) cannot contain additional positive or negative elements without falling into one of those categories. The "Between" and "Cycle" cases are similarly terse, relying on "directly check" and "as in previous cases" without providing the necessary arguments. For a classification theorem, these maximality proofs need to be written out in full.
[Section 4, Proposition 2 dependence] The completeness of the classification depends entirely on Proposition 2, cited from the authors' prior work [3], that Sym(Q) has exactly five closed subgroups containing the shift group. This is an external result, and the paper does not state it precisely or verify that its formulation matches the cited source. If Proposition 2 is incomplete or misquoted, the five-case split would miss possible spaces and the main theorem would fail. The authors should state explicitly which results from [1], [3], and [4] are being used in which step, and ensure the formulations align.
[Section 4, Lemma 3] The proof of the boundary lemma (every definable relation has a boundary) is only sketched. The argument uses a non-standard elementary extension and asserts that property (*) holds for a positive non-standard element m0, but the justification for this existence is not given. Since Lemma 3 is used in Lemma 4 to justify placing blocks at large distances, this step should be spelled out more carefully. The reference to Lemma 4.4 from [4] is helpful, but the proof as presented in the current paper is not self-contained enough for the key role it plays.

minor comments (7)

[Section 3] The definition of the relation A1n in Section 3 uses the difference x−y and absolute value |x−y| before the function "−" is defined in Section 4; consider moving the definition of subtraction to the preliminary section or adding a forward reference.
[Section 4, Lemma 1 proof] The text "the ratio R" should be "the relation R" in the sentence about checking conservation of relations.
[Section 4, "Equality" case] There is a typo: "comsideration" should be "consideration".
[Section 4, "Separation" case] The phrase "g+ decreases on some infinite element of the section" is unclear; it likely means "on an infinite part of the section" or "on some element of an infinite component of the section." The intended meaning should be clarified.
[Section 4, "Equality" case] The reference to "statement 2" in the sentence "according to the statement 2, Γ∗ contains all negative permutations" appears to refer to Proposition 3, not to any statement numbered 2. The numbering of propositions should be made consistent.
[Abstract and Section 3] The abstract mentions that the lattice is generated by relations including "equality", but the main theorem's interval [A11,<] does not explicitly mention equality. Please clarify how equality fits into the diagram and the statement of the main theorem.
[Figure 2] The diagram is not reproduced in the text provided; since the main theorem refers to "the Diagram (Fig.2)", the final manuscript must ensure the figure clearly labels all vertices and edges and that the inclusion claims are visually verifiable.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the interval classification is a genuine lifting of independent background results; the only vulnerable step is a combinatorial exhaustiveness claim in Lemma 4, not a circular reduction.

full rationale

The main theorem is not obtained by defining the target interval in terms of the cited background lattices. Proposition 1 ([1], same authors) is a general Svenonius-type anti-isomorphism for upward complete structures; it does not assume the [A11,<] classification. Proposition 2 ([3]) and Proposition 4 ([5]) are classifications of ⟨Q,<⟩ automorphism groups, again independent of the target interval. Lemma 3 is proved in the paper, and Lemma 1 is proved directly. The inference from the five initiated subgroups of Sym(Q) to the spaces in the Diagram uses Lemma 4; Lemma 4's proof rests on the assertion that the displayed block rearrangements 'can obtain every predetermined order of blocks' (Section 4, Lemma 4). That assertion is not shown in detail and is a genuine proof gap, but it is an unproved combinatorial claim, not a circular definition or a fitted input renamed as prediction. No equation in the paper reduces the target theorem to its own assumptions; the dependencies are external benchmarks. Hence no significant circularity; the self-citations are load-bearing but independent support.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters appear because the paper is a pure classification theorem with no fitted data. The central proof relies on the authors' own earlier theorems for the Svenonius correspondence, the rational order lattice, and the successor lattice, plus an in-paper boundary lemma. No new mathematical entities are postulated.

assumptions (5)

standard math Svenonius theorem for upward complete structures (Proposition 1)
Cited from the authors' own prior work [1]; establishes the anti-isomorphism between the definability lattice and the lattice of closed supergroups on Q times Z.
domain assumption Q times Z is the upward complete extension of <Z,<> (Section 2)
The proof shifts definability questions from <Z,<> to automorphism groups of Q times Z.
standard math The five-group classification of Sym(Q) (Proposition 2)
Cited from the authors' prior work [3]; the five-case analysis depends on this list being complete.
domain assumption The boundary lemma for definable relations (Lemma 3)
Proved in the paper via non-standard extension; it underpins Lemma 4's block permutation arguments.
standard math Description of Gamma(<Q,S>) (Proposition 4)
Cited from the authors' prior work [5]; used in the Separation case.

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Cite this review

Pith. "Pith review of On a lattice of relational spaces (reducts) for the order of integers." pith.science (2026). https://pith.science/paper/FF5MKRP2

@misc{pith2026241118181,
  author       = {Pith},
  title        = {Pith review of: On a lattice of relational spaces (reducts) for the order of integers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FF5MKRP2}},
  note         = {Machine review of arXiv:2411.18181}
}

read the original abstract

We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are proposed.

Figures

Figures reproduced from arXiv: 2411.18181 by the authors.

**Figure 1.** Lattice for ⟨Q, <⟩ Of course, for some linearly ordered sets, elements of this lattice may coincide. However, for example, for rational numbers it is not difficult to prove that all elements are different. 1 arXiv:2411.18181v1 [math.LO] 27 Nov 2024 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** The Diagram The main result of the paper is [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Definable coordinate geometries over fields, part 2: applications
math.LO 2025-07 accept novelty 6.0 of 10

Using an automorphism-group criterion, the authors determine exactly how the definable relations of ordered affine, Euclidean, Galilean, Newtonian, Relativistic, Minkowski, and Late Classical geometries are nested.
Definable coordinate geometries over fields, part 1: theory
math.LO 2025-07 accept novelty 6.0 of 10

For finitely field-definable coordinate geometries over ordered fields or fields with more than two elements, a relation is a concept if and only if it is field-definable and invariant under affine automorphisms, so c...

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 2 Pith papers

[1]

Automorphisms and Definability (of Reducts) for Upward Com- plete Structures

Semenov, A., Soprunov, S. Automorphisms and Definability (of Reducts) for Upward Com- plete Structures. Mathematics. 2022, 10(20), 3748. https://doi.org/10.3390/math10203748

work page doi:10.3390/math10203748 2022
[3]

A, Semenov, A

Muchnik, An. A, Semenov, A. L. Lattice of definability in the order of rational numbers. Mathematical Notes. 2020, 108, 94–107. https://doi.org/10.1134/S0001434620070093

work page doi:10.1134/s0001434620070093 2020
[4]

L.; Soprunov, S

Semenov, A. L.; Soprunov, S. F. Lattice of Definability (of Reducts) for Integers with Suc- cessor. Izvestiya: Mathematics 2021, 85:6, 1257–1269. https://doi.org/10.1070/IM9107

work page doi:10.1070/im9107 2021
[2]

Huntington, E. V. Inter-relations among the four principal types of order. Transactions of the American Mathematical Society.1935, 38(1), 1–9

work page 1935
[5]

Mappings preserving relations definable through linear order

Semenov, A.L. Mappings preserving relations definable through linear order. In Russian. Bulletin of Moscow University. Series 1. Mathematics. Mechanics, 2020, (5), pp. 62–65 (in Russian)

work page 2020

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Automorphisms and Definability (of Reducts) for Upward Com- plete Structures

Semenov, A., Soprunov, S. Automorphisms and Definability (of Reducts) for Upward Com- plete Structures. Mathematics. 2022, 10(20), 3748. https://doi.org/10.3390/math10203748

work page doi:10.3390/math10203748 2022

[2] [3]

A, Semenov, A

Muchnik, An. A, Semenov, A. L. Lattice of definability in the order of rational numbers. Mathematical Notes. 2020, 108, 94–107. https://doi.org/10.1134/S0001434620070093

work page doi:10.1134/s0001434620070093 2020

[3] [4]

L.; Soprunov, S

Semenov, A. L.; Soprunov, S. F. Lattice of Definability (of Reducts) for Integers with Suc- cessor. Izvestiya: Mathematics 2021, 85:6, 1257–1269. https://doi.org/10.1070/IM9107

work page doi:10.1070/im9107 2021

[4] [2]

Huntington, E. V. Inter-relations among the four principal types of order. Transactions of the American Mathematical Society.1935, 38(1), 1–9

work page 1935

[5] [5]

Mappings preserving relations definable through linear order

Semenov, A.L. Mappings preserving relations definable through linear order. In Russian. Bulletin of Moscow University. Series 1. Mathematics. Mechanics, 2020, (5), pp. 62–65 (in Russian)

work page 2020