arxiv: 2604.03080 · v2 · submitted 2026-04-03 · 🧮 math.LO · math.GR

Recognition: no theorem link

An unstable abstract elementary class of modules: A variation of Paolini-Shelah's example

Daniel Herden , Marcos Mazari-Armida , Michael D. Walton

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:17 UTC · model grok-4.3

classification 🧮 math.LO math.GR MSC 03C48

keywords abstract elementary classestorsion-free abelian groupsamalgamation propertystabilitytamenessjoint embedding propertypure subgroups

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The pith

A variation of a known construction produces an unstable AEC of torsion-free abelian groups that has joint embedding but lacks amalgamation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a class of torsion-free abelian groups ordered by pure subgroups that forms an abstract elementary class with countable Löwenheim-Skolem number. This class fails stability, satisfies the joint embedding property, admits no maximal models, lacks the amalgamation property, and satisfies countable tameness. The authors obtain the example by modifying the Paolini-Shelah construction so that only the listed combination of properties survives. A reader would care because the example separates stability, amalgamation, and tameness inside a concrete category of modules.

Core claim

We construct a class hat K of torsion-free abelian groups such that hat K = (hat K, ≤_p) is an abstract elementary class with LS(hat K) = ℵ₀ such that hat K is not stable, hat K has the joint embedding property and no maximal models but does not have the amalgamation property, and hat K is (<ℵ₀)-tame. The construction is obtained by varying the example from Paolini-Shelah Section 4 in order to isolate the core mechanism responsible for these features.

What carries the argument

The class hat K of torsion-free abelian groups equipped with the partial order ≤_p of pure subgroups, obtained via a targeted variation of the Paolini-Shelah construction.

If this is right

The class satisfies the joint embedding property and has no maximal models.
The class fails to be stable.
The class fails the amalgamation property.
The class is (<ℵ₀)-tame.
The Löwenheim-Skolem number of the class equals ℵ₀.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same variation technique may be applied to other module categories to produce further examples that separate tameness from amalgamation.
The construction supplies a concrete test case for whether countable tameness plus joint embedding can force any form of local character in AECs of modules.
One could check whether the class satisfies additional properties such as having a model-theoretic independence relation that is not forking.

Load-bearing premise

The specific variation performed in Section 4 removes stability and amalgamation while preserving joint embedding, the absence of maximal models, and countable tameness.

What would settle it

An explicit pair of models in hat K together with a common submodel that cannot be amalgamated inside hat K, or a demonstration that the class satisfies the stability definition.

read the original abstract

We construct a class $\hat{K}$ of torsion-free abelian groups such that $\hat{\mathbf{K}}=(\hat{K}, \leq_p)$ is an abstract elementary class with $\operatorname{LS}(\hat{\mathbf{K}})=\aleph_0$ such that: $(\cdot)$ $\hat{\mathbf{K}}$ is not stable; $(\cdot)$ $\hat{\mathbf{K}}$ has the joint embedding property and no maximal models, but does not have the amalgamation property; $(\cdot)$ $\hat{\mathbf{K}}$ is $(<\aleph_0)$-tame. The class we construct is a variation of [PaSh, Section 4] which isolates the core mechanism of the Paolini-Shelah construction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A useful variation that isolates an unstable tame AEC of torsion-free groups with JEP but no AP.

read the letter

The main takeaway is that this paper produces a concrete AEC of torsion-free abelian groups that is unstable, has the joint embedding property without amalgamation or maximal models, and is tame for countable sets, by adjusting the Paolini-Shelah construction to separate out the key pieces more clearly. The class uses a purity relation on the groups to satisfy the AEC axioms with Löwenheim-Skolem number aleph zero. They show instability by exhibiting splitting types, give explicit diagrams of groups that cannot be amalgamated inside the class, and get tameness from the fact that the relations depend only on countable data. This makes the example more transparent than the original and supplies a specific data point for questions about which bundles of properties can coexist in module classes. The construction looks internally consistent, with no obvious extra assumptions that would force amalgamation or stability back in. The main limitation is that the work is presented as a variation rather than a new general technique, so the advance is mostly in sharpening an existing idea and verifying that the listed properties survive the tweak. Citations stay focused on the relevant prior work without unnecessary padding. This is aimed at people working on classification theory for abstract elementary classes, especially those who track examples in algebra to test broader claims about stability and amalgamation. A reader who needs a verifiable instance of these properties together will get something out of it. The details are explicit enough that the claims can be checked, so the paper deserves a serious referee even though the step forward is incremental rather than foundational. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs a class hat K of torsion-free abelian groups such that hat K = (hat K, ≤_p) forms an abstract elementary class with LS(hat K) = ℵ₀. It establishes that the AEC is unstable, satisfies the joint embedding property and has no maximal models, fails the amalgamation property, and is (<ℵ₀)-tame. The construction is a targeted variation of the Paolini-Shelah example from their Section 4, intended to isolate the mechanism producing these properties.

Significance. If the proofs are correct, the example supplies a concrete, tame unstable AEC of countable Löwenheim-Skolem number that has JEP but lacks AP. Such an example is useful for mapping the independence of stability, amalgamation, and tameness in AECs and for providing an algebraic (module-theoretic) setting in which these failures can be examined directly.

major comments (2)

[§4] §4: The variation of the purity relation ≤_p is defined by modifying the original Paolini-Shelah parameters, but the verification that this modification preserves the AEC axioms (in particular coherence and the smoothness property) is only sketched; an explicit check that the new relation satisfies the chain-union condition for directed systems would confirm that the central claim of being an AEC is not affected by the change.
[§5.1] §5.1: The explicit non-amalgamable diagram witnessing failure of AP is presented, yet the argument that no amalgam exists inside hat K relies on the specific form of ≤_p; a short calculation showing that any candidate amalgam would violate the purity condition on at least one of the embeddings would make the non-amalgamation claim fully load-bearing and self-contained.

minor comments (2)

The notation switches between hat K and boldface K in the abstract and early sections; uniform use of one convention throughout would reduce minor confusion.
[Introduction] A brief sentence in the introduction recalling the precise statement of the original Paolini-Shelah example (rather than only citing the section) would help readers see exactly which features were isolated by the variation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the careful reading of our manuscript and for the helpful suggestions. We address each major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [§4] §4: The variation of the purity relation ≤_p is defined by modifying the original Paolini-Shelah parameters, but the verification that this modification preserves the AEC axioms (in particular coherence and the smoothness property) is only sketched; an explicit check that the new relation satisfies the chain-union condition for directed systems would confirm that the central claim of being an AEC is not affected by the change.

Authors: We agree that an explicit verification of the chain-union condition would strengthen the presentation. In the revised manuscript, we will include a detailed calculation showing that the modified ≤_p satisfies the required properties for directed systems, adapting the argument from the original Paolini-Shelah paper to our parameter choices. This will confirm that the AEC axioms hold without relying solely on the sketch. revision: yes
Referee: [§5.1] §5.1: The explicit non-amalgamable diagram witnessing failure of AP is presented, yet the argument that no amalgam exists inside hat K relies on the specific form of ≤_p; a short calculation showing that any candidate amalgam would violate the purity condition on at least one of the embeddings would make the non-amalgamation claim fully load-bearing and self-contained.

Authors: Thank you for this suggestion. We will add a short explicit calculation in §5.1 demonstrating that any potential amalgam in hat K would necessarily violate the purity condition ≤_p for at least one of the embeddings. This will make the failure of amalgamation self-contained and independent of external references. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper defines the class hat K directly via a variation on the external Paolini-Shelah construction and verifies the AEC axioms, non-stability, JEP without AP, and (<aleph0)-tameness through explicit definitions of the purity relation ≤_p together with concrete counterexamples for the failure of amalgamation and stability. No parameters are fitted, no equations reduce the target properties to the inputs by construction, and the single reference to prior work is not a self-citation load-bearing step. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard definition of an AEC and the existence of a suitable variation of the cited construction; no free parameters are introduced in the abstract, and the new class is the main invented entity.

axioms (1)

domain assumption The structure (hat K, ≤_p) satisfies the axioms of an abstract elementary class with LS number aleph0
Invoked directly in the statement of the main result.

invented entities (1)

hat K class of torsion-free abelian groups with ≤_p no independent evidence
purpose: To realize an AEC with the listed stability, embedding, and tameness properties
The paper constructs this class as the central object.

pith-pipeline@v0.9.0 · 5429 in / 1389 out tokens · 25317 ms · 2026-05-13T18:17:50.713105+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 1 internal anchor

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