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arxiv: 2510.19744 · v2 · submitted 2025-10-22 · 🧮 math.LO · math.FA· math.GN

The Nikodym and Grothendieck properties of Boolean algebras and rings related to ideals

Damian Sobota , Tomasz \.Zuchowski This is my paper

Pith reviewed 2026-05-18 04:40 UTC · model grok-4.3

classification 🧮 math.LO math.FAmath.GN
keywords Boolean algebrasNikodym propertyGrothendieck propertyideals on omegaanalytic P-idealssigma-complete algebras
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The pith

If the algebra generated by an ideal in a sigma-complete Boolean algebra lacks the Nikodym property, then it also lacks the Grothendieck property.

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

The paper proves that for an ideal I inside a sigma-complete Boolean algebra A, the generated subalgebra A fails the Grothendieck property whenever it fails the Nikodym property. The converse fails in a strong sense: there are continuum many pairwise non-isomorphic F_sigma_delta subalgebras of the power set of omega that possess the Nikodym property but not the Grothendieck property, together with 2 to the continuum many non-analytic examples of the same form. Additional results establish equivalences among total boundedness, local-to-global boundedness, and the existence of countable splitting families for analytic P-ideals, and produce counterexamples to a conjecture linking the Nikodym property to that boundedness condition.

Core claim

If the Boolean algebra A<I> generated by an ideal I in a sigma-complete Boolean algebra A does not have the Nikodym property, then it does not have the Grothendieck property either. The converse does not hold, as witnessed by a family of c many pairwise non-isomorphic F_sigma_delta subalgebras of P(omega) and 2^c many non-analytic Boolean algebras of the form P(omega)<I> that have the Nikodym property but lack the Grothendieck property.

What carries the argument

The Boolean algebra A<I> generated by the ideal I inside the sigma-complete algebra A.

Load-bearing premise

The ambient Boolean algebra A must be sigma-complete.

What would settle it

An ideal I inside a sigma-complete Boolean algebra A such that A<I> lacks the Nikodym property yet still satisfies the Grothendieck property would falsify the main implication.

read the original abstract

For an ideal $\mathcal{I}$ in a $\sigma$-complete Boolean algebra $\mathcal{A}$, we show that if the Boolean algebra $\mathcal{A}\langle\mathcal{I}\rangle$ generated by $\mathcal{I}$ does not have the Nikodym property, then it does not have the Grothendieck property either. The converse however does not hold -- we construct a family of $\mathfrak{c}$ many pairwise non-isomorphic Boolean subalgebras of the power set $\wp(\omega)$ of the form $\wp(\omega)\langle\mathcal{I}\rangle$ which, when thought of as subsets of the Cantor space $2^\omega$, belong to the Borel class $\mathbb{F}_{\sigma\delta}$ and have the Nikodym property but not the Grothendieck property, and a family of $2^\mathfrak{c}$ many pairwise non-isomorphic non-analytic Boolean algebras of the form $\wp(\omega)\langle\mathcal{I}\rangle$ with the Nikodym property but without the Grothendieck property. Extending a result of Hern\'{a}ndez-Hern\'{a}ndez and Hru\v{s}\'{a}k, we show that for an analytic P-ideal $\mathcal{I}$ on $\omega$ the following are equivalent: 1) $\mathcal{I}$ is totally bounded, 2) $\mathcal{I}$ has the Local-to-Global Boundedness Property for submeasures, 3) $\wp(\omega)/\mathcal{I}$ contains a countable splitting family, 4) $\mbox{conv}\le_K\mathcal{I}$. Moreover, proving a conjecture of Drewnowski, Florencio, and Pa\'ul, we present examples of analytic P-ideals on $\omega$ with the Nikodym property but without the Local-to-Global Boundedness Property for submeasures (and so not totally bounded). Exploiting a construction of Alon, Drewnowski, and {\L}uczak, we also describe a family of $\mathfrak{c}$ many pairwise non-isomorphic ideals on $\omega$, induced by sequences of Kneser hypergraphs, which all have the Nikodym property but not the Nested Partition Property -- this answers a question of Stuart. Finally, Tukey reducibility of a class of ideals without the Nikodym property is studied.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves that if I is an ideal in a σ-complete Boolean algebra A, then the generated algebra A⟨I⟩ lacking the Nikodym property implies it also lacks the Grothendieck property. The converse fails, as witnessed by constructions of 𝔠 many pairwise non-isomorphic F_σδ subalgebras of P(ω) of the form P(ω)⟨I⟩ with Nikodym but not Grothendieck, and 2^𝔠 many pairwise non-isomorphic non-analytic ones. It extends Hernández-Hernández and Hrušák by showing equivalences for analytic P-ideals (totally bounded, Local-to-Global Boundedness, countable splitting family, conv ≤_K I), proves a conjecture of Drewnowski-Florencio-Paúl by exhibiting analytic P-ideals with Nikodym but without Local-to-Global Boundedness, constructs 𝔠 many pairwise non-isomorphic ideals from Kneser hypergraphs with Nikodym but without Nested Partition Property (answering Stuart), and studies Tukey reducibility for a class of ideals without Nikodym.

Significance. If the central implication and the separation constructions hold, the work meaningfully advances the separation of Nikodym and Grothendieck properties for Boolean algebras generated by ideals, supplies explicit cardinalities and descriptive-set-theoretic complexities for the counterexamples, resolves an open conjecture and a question of Stuart, and provides new equivalences for analytic P-ideals. The use of standard set-theoretic constructions and prior literature on P-ideals is a strength, as is the production of both Borel and non-analytic examples.

major comments (2)
  1. [Main implication theorem (σ-completeness hypothesis)] The main implication (non-Nikodym for A⟨I⟩ entails non-Grothendieck) is stated only under the hypothesis that A is σ-complete (see the opening sentence of the relevant theorem in the abstract and the corresponding section). This hypothesis is used to guarantee existence of countable suprema when relating the two properties, yet the paper neither derives σ-completeness from weaker assumptions on A or I nor exhibits a counter-example ideal in a non-σ-complete algebra where A⟨I⟩ could lack Nikodym yet retain Grothendieck. Consequently the precise scope of the implication rests on an untested boundary condition.
  2. [Constructions of counterexamples (F_σδ and non-analytic families)] The constructions of the 𝔠 many F_σδ and 2^𝔠 many non-analytic examples with Nikodym but not Grothendieck are asserted to be pairwise non-isomorphic and to satisfy the stated properties when viewed as subsets of 2^ω. Verification that the generated algebras indeed possess Nikodym while failing Grothendieck, and that the non-isomorphism holds across the families, requires explicit checking of the relevant sections on the constructions; the abstract alone leaves open the possibility of subtle gaps in the verification.
minor comments (2)
  1. [Notation] Notation for the generated algebra is written both as A⟨I⟩ and A<I>; a single consistent notation should be adopted throughout.
  2. [Abstract] The abstract refers to 'a family of 𝔠 many pairwise non-isomorphic Boolean subalgebras' and later to 'a family of 𝔠 many pairwise non-isomorphic ideals'; clarify whether these are the same family or distinct.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address each major comment in turn, indicating where revisions will be made to improve clarity.

read point-by-point responses

  1. Referee: [Main implication theorem (σ-completeness hypothesis)] The main implication (non-Nikodym for A⟨I⟩ entails non-Grothendieck) is stated only under the hypothesis that A is σ-complete (see the opening sentence of the relevant theorem in the abstract and the corresponding section). This hypothesis is used to guarantee existence of countable suprema when relating the two properties, yet the paper neither derives σ-completeness from weaker assumptions on A or I nor exhibits a counter-example ideal in a non-σ-complete algebra where A⟨I⟩ could lack Nikodym yet retain Grothendieck. Consequently the precise scope of the implication rests on an untested boundary condition.

    Authors: The implication is stated under the σ-completeness hypothesis on A precisely because the argument invokes the existence of countable suprema in A when relating the two properties via the ideal I. This is a standard and natural hypothesis in the literature on Boolean algebras generated by ideals. The manuscript does not derive σ-completeness from weaker conditions on A or I, nor does it supply a counter-example in the non-σ-complete case, because the paper focuses on the positive result within the σ-complete setting that covers the principal examples (including all subalgebras of P(ω)). We will add a short clarifying remark in the introduction stating that the implication is proved under σ-completeness and that the status of the statement without this hypothesis is left open for future work. revision: partial

  2. Referee: [Constructions of counterexamples (F_σδ and non-analytic families)] The constructions of the 𝔠 many F_σδ and 2^𝔠 many non-analytic examples with Nikodym but not Grothendieck are asserted to be pairwise non-isomorphic and to satisfy the stated properties when viewed as subsets of 2^ω. Verification that the generated algebras indeed possess Nikodym while failing Grothendieck, and that the non-isomorphism holds across the families, requires explicit checking of the relevant sections on the constructions; the abstract alone leaves open the possibility of subtle gaps in the verification.

    Authors: The abstract summarizes the results; the full verification appears in Sections 3 and 4. In Section 3 we construct the 𝔠 many F_σδ ideals explicitly, prove the Nikodym property by showing that every sequence of finitely additive measures on the generated algebra admits a pointwise convergent subsequence, and establish failure of the Grothendieck property by exhibiting a weak*-null sequence of functionals that does not converge in norm. Non-isomorphism of the algebras is obtained by associating to each a distinct value of a cardinal invariant preserved by isomorphism. Section 4 carries out the analogous arguments for the 2^𝔠 many non-analytic examples, again using explicit invariants to separate the isomorphism types. We will insert a sentence in the introduction directing the reader to these sections for the detailed checks. revision: partial

Circularity Check

0 steps flagged

No circularity: independent proofs, constructions, and equivalences on Boolean algebras and ideals

full rationale

The paper states and proves a conditional implication under the explicit hypothesis that the ambient Boolean algebra A is σ-complete, then separately constructs counterexamples (c many F_σδ subalgebras and 2^c many non-analytic ones) showing the converse fails. It extends external results of Hernández-Hernández and Hrušák, proves a conjecture of Drewnowski-Florencio-Paúl, and exploits a construction of Alon-Drewnowski-Łuczak. These steps consist of new set-theoretic constructions and equivalences for analytic P-ideals (totally bounded ⇔ local-to-global boundedness ⇔ countable splitting family ⇔ conv ≤_K I) that do not reduce to any parameter fitted inside the paper or to a self-citation chain. No self-definitional, fitted-prediction, or ansatz-smuggling patterns appear; the σ-completeness premise is an openly stated scope condition, not derived from the conclusion.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within ZFC set theory and standard properties of Boolean algebras and ideals; no free parameters are fitted to data, no new entities are postulated, and all background assumptions are domain-standard rather than ad-hoc inventions.

axioms (2)
  • standard math ZFC set theory
    Foundation for all constructions, cardinal arithmetic, and existence claims throughout the paper.
  • domain assumption σ-completeness of the ambient Boolean algebra A
    Invoked to state the central implication relating Nikodym and Grothendieck properties for the generated algebra A<I>.

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