Extending orders to types

Boris \v{S}obot; Lorenzo Luperi Baglini; Marcello Mamino; Mariaclara Ragosta; Rosario Mennuni

REVIEW 3 major objections 4 minor 14 references

Extending orders to types

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read For definably complete linear orders, the order-induced preorder on 1-types is isomorphic to the ordered space of cuts in the definable closure, and this transfer to divisibility on ultrafilters yields an independence result from ZFC.

desk verdict A clean model-theoretic characterization of 1-type spaces for definably complete orders, with a nice ZFC-independence payoff for the Ep orders; the paper deserves refereeing, though the finite-case classification needs an exhaustiveness argument. read the letter →

arxiv 2509.09623 v2 pith:YU6PCV6O submitted 2025-09-11 math.LO

Lorenzo Luperi Baglini , Marcello Mamino , Rosario Mennuni , Mariaclara Ragosta , Boris Šobot This is my paper

classification math.LO MSC 03C6403H1554D80

keywords order-inducedpreorderstypespacesdefinablecompletenesscutsinlinearordersultrafiltersonNdivisibilitypreorderprimenonstandardintegers

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 gives a complete description of the preorder that an order induces on the space of 1-types over a parameter set, for structures that are definably complete linear orders. It proves that two 1-types are mutually comparable under this preorder exactly when they determine the same cut in the definable closure of the parameters, so the quotient order is isomorphic to the space of such cuts. It then transfers this result to the divisibility preorder on ultrafilters on the natural numbers: for a prime ultrafilter p, the suborder of ultrafilters whose only prime divisor has type p is isomorphic to the cut-space of a generator of p. This yields an independence result: under the continuum hypothesis all nonprincipal such suborders are isomorphic, while in certain models of not-CH they are not. The paper also classifies the points of the analogous posets for ultrafilters with finitely many prime divisors into five kinds.

What carries the argument

The central mechanism is the cut of a type: for p ∈ S1(A), Lp = {a ∈ dcl(A) : p ⊢ x ≥ a} and Rp = {a ∈ dcl(A) : p ⊢ x ≤ a}, whose type-definable set is the cut. Definable completeness supplies the key property that p is realised cofinally and coinitially inside that cut (Lemma 3.6), which makes the cut-to-class map injective; the natural linear order on cuts then mirrors the preorder ≾. In the divisibility application, the bridge is Remark 4.5: the map sending the =∼-class of tp(γη/N) to the ≈-class of tp(η/N(γ)) is an isomorphism between Ep and S1(γ)/≈. Lemma 4.9, identifying the order type of N(A) as N followed by copies of Z and cuts realised in dcl(A) with those having an immediate succe

What would settle it

Exhibit a definably complete linear order M, a small parameter set A, and two 1-types p and q over A that have the same cut in dcl(A) but for which no realisations α |= p, β |= q satisfy α ≤ β ≤ α′ for some α′ |= p; that would break Theorem 3.10. For the independence theorem, check Lemma 4.9 directly by computing the order type of N(γ) for a nonprincipal γ and looking for a cut realised in dcl(γ) whose successor cut is not realised — finding one would block the cofinality transfer in Theorem 4.10(b).

Watch

Extended reading notes

Core claim

The main theorem (Theorem 3.10) states that for a definably complete expansion of a linear order, the natural map from S1(A)/≈ — 1-types up to mutual comparability under the order-induced preorder — to CC(A), the space of cuts in the definable closure of A that are realised in some elementary extension, is an isomorphism of linear orders. The proof hinges on Lemma 3.6, which shows that every 1-type is realised coinitially and cofinally many times inside the set of realisations of its cut; this makes the cut determine the ≈-class. Applied to the full structure on N with divisibility, this gives Ep ≅ CC(γ) for any generator γ of a prime ultrafilter p, reducing the order type of the divisibilit

Load-bearing premise

The argument leans on Lemma 4.9's unproved description of the order type of N(A) as N followed by copies of Z, with a cut in dcl(A) realised exactly when it has an immediate successor in CC(A), and on the stated-without-proof exhaustiveness of the five-way classification of points of Eq.

Editorial extensions

If this is right

For every prime ultrafilter p with generator γ, the divisibility suborder Ep is isomorphic to the cut space CC(γ); hence Rudin–Keisler reducibility of p to p′ embeds Ep into Ep′, while Rudin–Keisler equivalence makes them isomorphic.
Under the continuum hypothesis, all Ep with p nonprincipal are isomorphic; in a model of not-CH expanded by at least continuum many Cohen reals, there exist nonprincipal p and q with Ep not isomorphic to Eq.
If p is a standard prime, then Ep is isomorphic to ω + 1.
For a type q of an increasing k-tuple of primes, Eq is isomorphic to (Sk(γ), ≾)/≈, and every point of Eq falls into one of five mutually exclusive kinds (a)–(e).
The higher-dimensional analogue of the main cut-space theorem fails: Example 3.11 gives two 2-types with the same quantifier-free ≤-type but different ≈-classes.

Reading between the lines

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

The cut-space isomorphism makes the order type of Ep a purely order-theoretic invariant of the ultrapower N(γ); one could try to recover properties of the ultrafilter p from first-order or cofinality properties of CC(γ), potentially distinguishing more ultrafilters than Rudin–Keisler equivalence does.
The five-kind classification suggests that for k ≥ 2 the right invariant is not a single cut but the arrangement of coordinatewise cuts together with the antichain-versus-convex-locus dichotomy; testing this on tensor pairs may yield a finer invariant for Eq.
If Lemma 4.9 is proved in full, the independence argument shows that the isomorphism type of Ep is sensitive to the cofinality of N^N/p; examining whether CC(γ) contains a canonical definable copy of N(γ) could provide a way to compute that cofinality directly.
The squarefree-divisibility examples in Section 5 suggest that the cut-based approach might extend to the infinite-prime-divisor case once single generators are replaced by patterns, since the antichain criterion for singleton classes already transfers.

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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

Dear [Colleague],

You should know about this paper because it settles the isomorphism question for the Ep orders and does so with a clean model-theoretic theorem. The main result (Theorem 3.10) is the real content: for a definably complete linear order, the natural preorder on 1-types over A, taken modulo mutual comparability, is isomorphic to the linearly ordered set of cuts in dcl(A) that are filled in some elementary extension. The proof is short and standard—saturation for surjectivity, compactness plus definable completeness for injectivity—and Example 3.7 shows the definable completeness assumption is needed. The authors are honest that the preorder itself already appears as Definition 1.3 in PS21; what is new is this cut characterization and its consequences.

The ultrafilter applications are a nice payoff. Theorem 4.6 identifies the order Ep with the cut space over a generator γ, and Theorem 4.10 shows (via Roitman's theorem) that the isomorphism type of Ep for nonprincipal p is independent of ZFC: under CH they're all isomorphic, while after adding enough Cohen reals they're not. That directly answers a question from Šobot. The classification of points in Eq into five kinds is useful and comes with examples, though as I say below it needs a bit more care.

Where are the soft spots? Lemma 4.9 is used to transfer cofinality differences from ultrapowers to non-isomorphism of Ep, and its proof is compressed: it asserts without proof that dcl(A) has order type N followed by copies of Z. That is true for models of arithmetic, and any model theorist will fill in the details, but it should be stated and justified explicitly. The bigger issue is the five-case classification in Section 4.2. The authors list five mutually exclusive cases and illustrate each, but they never prove exhaustiveness—that every point of Eq falls into one of them. That feels like a real gap in the presentation, even if the claim is plausible, and it should not be left as an exercise. Neither issue touches Theorem 3.10, which I'm convinced is correct.

This is worthwhile work for anyone in model theory of orders or ultrafilter theory. It deserves a serious referee. If I were the editor, I'd send it out and ask the authors to expand the proof of Lemma 4.9 and add an explicit argument that the five cases cover everything.

Best.

Referee Report

3 major / 4 minor

Summary. The paper defines a preorder ≾ on type spaces over ordered structures: p ≾ q iff some realisations α |= p, β |= q satisfy α ≤ β in the product order. For linearly ordered, definably complete structures the central Theorem 3.10 identifies the quotient S1(A)/≈ with the order CC(A) of consistent cuts in dcl(A). The proof is short, uses compactness and saturation, and is complemented by an example showing definable completeness is necessary. The paper then applies this to the divisibility preorder on βN, proving that for a prime ultrafilter p the poset Ep of its powers is isomorphic to CC(γ) for γ |= p, establishing an independence result for the isomorphism type of Ep under CH versus ¬CH via Roitman's theorem, and offering a five-case classification of ultrafilters with finitely many prime divisors.

Significance. If the main theorem and its applications hold, the paper gives a clean model-theoretic description of 1-type preorders in definably complete linear orders and connects it with the ultrafilter divisibility order in a non-obvious way. The proof of Theorem 3.10 is genuinely elegant, and the use of Lemma 3.6 to show that each type is coinitial and cofinal in its cut is a nice observation. The paper also makes good use of standard saturation and of Roitman's theorem for the independence result. However, the ultrafilter applications rely on several claims that are asserted rather than proved (notably Remark 4.5 and Lemma 4.9), and the five-case classification in §4.2 has no stated exhaustiveness proof. The central model-theoretic theorem is sound, but the advertised applications need additional support before the paper can be accepted in its current form.

major comments (3)

[§4.1, Remark 4.5 and Theorem 4.6] The isomorphism Ep ≅ S1(γ)/≈ is the bridge between the main theorem and the ultrafilter applications, but its proof is only a sketch. Three things need to be established: (i) every =∼-class in Ep has a representative of the form γ^η with the same fixed generator γ; (ii) divisibility between p-powers is witnessed by a common generator with ordered exponents; and (iii) the map is order-preserving and bijective on classes. The sentence 'there must be γ |= p and η0 ≤ η1' after changing generators is not justified. Since Theorem 4.6 and the later independence theorem rely on this correspondence, a proof or a precise citation to [Šob25a] or [Šob25b] is necessary.
[§4.1, Lemma 4.9] The proof of Lemma 4.9 is one sentence: 'Because dcl(A)=N(A) is a model, its order type is that of N followed by copies of Z.' This is not immediate. Nonstandard models of arithmetic have order type ω + ζ·η with η dense, and the behaviour of cuts in CC(A) depends on the ordering of the Z-chains. The equivalence 'realised in dcl(A) iff it has an immediate successor' is exactly what transfers cofinality differences of ultrapowers in Theorem 4.10(b), so this lemma is load-bearing. Please give a proof or a precise reference for both the order-type statement and the cut characterisation.
[§4.2, cases (a)–(e)] The five-case classification is stated as 'five mutually exclusive cases' but no exhaustiveness proof is supplied. In particular, after excluding cases (a), (b), and (c), it is not shown why every remaining non-singleton class falls into case (d), nor how the reduction in case (e) handles a mixture of coordinates in N(γ) and coordinates not in N(γ) when the latter are not all of the same kind. The classification is advertised in the abstract, so a proof that the cases cover all possibilities is required.

minor comments (4)

[§4.1, Theorem 4.10(b)] The line 'Let γi |= Epℵi' should presumably read 'Let γi |= pℵi'; as written, Epℵi is a poset, not a type.
[§3, Example 3.7] The example is terse. It would help to state explicitly that definable completeness fails because P(Q) is bounded above but has no supremum, and to explain how the ¬P part of the cut prevents p from being cofinal.
[§4.2, Remark 4.15(b)] The notion of 'supremum' in this context is not defined. Specify the order in which the supremum is taken (e.g., in the quotient S_k(γ)/≈ or in E_q).
[§2.1, Definition 2.1] The relation is denoted ⪅ in Definition 2.1 and ≾ elsewhere; the notation should be unified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: main theorem proved from definitions; self-citations are motivational only.

full rationale

The derivation is self-contained. Theorem 3.10 is proved from the definitions of the preorder, cuts, and definable closure via Lemma 3.6, whose proof uses definable completeness, compactness, and saturation. Injectivity of the cut map follows from the fact that each type is realized coinitially and cofinally in its cut; surjectivity follows from Remark 3.9, which identifies CC(A) with cuts realized in the monster model. No identifications are built into the definitions. The Section 4 applications reduce Ep to S1(γ)/≈ by an actual proof in Remark 4.5 that mutual divisibility of prime-power ultrafilters corresponds to mutual ≤-comparability of exponents over N(γ), and then apply Theorem 3.10. The ZFC-independence argument uses an external theorem of Roitman and Lemma 4.9, which is not circular: it states a standard property of the order type of models of arithmetic. Self-citations ([Šob21], [Šob25a], [LB19], [Šob25b]) appear only as motivation, open questions, or examples, not as load-bearing premises. The only rigor soft spots—Lemma 4.9's unproved order-type assertion and the unproved exhaustiveness of the five-case classification in §4.2—are standard or derivable claims, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new primitive entities; it defines new relations and orders on existing type spaces and ultrafilter spaces. The constructions are the preorder ≾, the equivalence ≈, the cut space CC(A), and the posets Ep/Eq. No free parameters are fitted; the only parameters are definitions. The axioms are standard set-theoretic and model-theoretic background plus the structural assumption of definable completeness and the full language on N.

assumptions (6)

standard math ZFC
All results are proved within ZFC; the independence proof presupposes consistency of ZFC.
standard math Existence of a κ-saturated and κ-strongly homogeneous monster model U for a large enough κ
Stated at the start of Section 2; used for compactness, saturation, and automorphism arguments throughout.
domain assumption The structure on N expands by symbols for every subset of each N^k (full language)
Assumed in Section 4 to identify k-types over N with ultrafilters in βN^k and to give N(A) Skolem functions.
domain assumption Definable completeness of the ordered structure (Assumption 3.3)
The main theorem is proved under this assumption; Example 3.7 shows it is necessary.
standard math Roitman's theorem on ultrapower cofinalities in Cohen forcing extensions
Cited in the proof of Theorem 4.10(b) to obtain ultrafilters pλ with cofinality λ; a published theorem not reproved.
domain assumption Order type of any N(A) is N followed by copies of Z; realized cuts in CC(A) are exactly those with an immediate successor
Lemma 4.9 states this and uses it for the independence proof; the order-type half is not proved in the paper.

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Pith. "Pith review of Extending orders to types." pith.science (2026). https://pith.science/paper/YU6PCV6O

@misc{pith2026250909623,
  author       = {Pith},
  title        = {Pith review of: Extending orders to types},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YU6PCV6O}},
  note         = {Machine review of arXiv:2509.09623}
}

read the original abstract

Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of 1-types. We apply these results to the divisibility preorder on the space of ultrafilters on the set of natural numbers, giving an independence result about the suborder consisting of ultrafilters with only one fixed prime divisor, as well as a classification of ultrafilters with finitely many prime divisors.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Countable ultraproducts without CH

Michael Canjar. Countable ultraproducts without CH . Ann. Pure Appl. Logic , 37(1):1--79, 1988

1988

[2] [2]

Self-divisible ultrafilters and congruences in Z

Mauro Di Nasso, Lorenzo Luperi Baglini, Rosario Mennuni, Moreno Pierobon, and Mariaclara Ragosta. Self-divisible ultrafilters and congruences in Z . J. Symb. Log. , 2023. doi: https://doi.org/10.1017/jsl.2023.51 10.1017/jsl.2023.51

doi:10.1017/jsl.2023.51 2023

[3] [3]

Definably complete Baire structures

Antongiulio Fornasiero and Tamara Servi. Definably complete Baire structures. Fundam. Math. , 209(3):215--241, 2010

2010

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An analogue of the Baire category theorem

Philipp Hieronymi. An analogue of the Baire category theorem. J. Symb. Log. , 78(1):207--213, 2013

2013

[5] [5]

Definability patterns and their symmetries

Ehud Hrushovski. Definability patterns and their symmetries. Preprint, https://arxiv.org/abs/1911.01129 arxiv.org/abs/1911.01129 , 2019

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Neil Hindman and Dona Strauss. Algebra in the Stone- Cech Compactification : Theory and Applications . De Gruyter, 2011

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Chris Miller. Expansions of dense linear orders with the intermediate value property. J. Symb. Log. , 66(4):1783--1790, 2001

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