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arxiv: 2606.09580 · v1 · pith:SAGBWPYKnew · submitted 2026-06-08 · 🧮 math.LO

A characterization of Projective and Weakly Projective Boolean Algebras

Andrzej Kucharski , S{\l}awomir Turek This is my paper

Pith reviewed 2026-06-27 14:11 UTC · model grok-4.3

classification 🧮 math.LO
keywords Boolean algebrasprojective Boolean algebrasweakly projective Boolean algebrasFreese-Nation propertycharacterizationprojectivityBoolean algebra theory
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The pith

Projective and weakly projective Boolean algebras are characterized by a modified version of the Freese-Nation property.

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

The paper aims to characterize projective and weakly projective Boolean algebras by introducing a modification of the Freese-Nation property. A sympathetic reader would care because standard definitions of projectivity rely on lifting properties or retracts, and an equivalent property expressed in terms of a Freese-Nation-style condition could simplify verification and classification. If the characterization holds, it supplies a uniform test that applies to both the projective and weakly projective cases. The work therefore replaces one set of algebraic definitions with another that may be easier to check in concrete examples.

Core claim

The paper establishes that projective Boolean algebras and weakly projective Boolean algebras can be precisely characterized using a suitable modification of the Freese-Nation property.

What carries the argument

A modification of the Freese-Nation property adapted to capture exactly the projective and weakly projective Boolean algebras.

Load-bearing premise

The proposed modification of the Freese-Nation property is defined so that it holds precisely for the projective and weakly projective Boolean algebras.

What would settle it

A single Boolean algebra that satisfies the modified property yet fails to be projective, or a projective Boolean algebra that fails the modified property.

read the original abstract

The aim of this paper is to give a characterization of projective and weakly projective Boolean algebras using some modification of the Freese-Nation property.

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

1 major / 0 minor

Summary. The paper aims to characterize projective and weakly projective Boolean algebras via a modification of the Freese-Nation property.

Significance. If a precise and verifiable modification of the Freese-Nation property were shown to capture exactly the projective and weakly projective cases, the result would supply a new criterion potentially useful to researchers working on Boolean algebras, their projective properties, and related questions in infinitary combinatorics and set theory. The Freese-Nation property is already a recognized tool in this area, so a successful modification could streamline certain classification arguments.

major comments (1)
  1. [Abstract] Abstract: The abstract states the aim of the paper but supplies neither a definition of the proposed modification of the Freese-Nation property, nor any statement of the main theorems, nor an outline of the proof strategy or verification steps. Without these elements the central claim cannot be evaluated for internal consistency or correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We agree that the abstract is minimal and will revise it to include the necessary details for evaluating the central claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The abstract states the aim of the paper but supplies neither a definition of the proposed modification of the Freese-Nation property, nor any statement of the main theorems, nor an outline of the proof strategy or verification steps. Without these elements the central claim cannot be evaluated for internal consistency or correctness.

    Authors: We accept this observation. The current abstract is deliberately concise but, as noted, omits key information. In the revised manuscript we will expand the abstract to (i) give a brief definition of the modified Freese-Nation property employed, (ii) state the two main characterization theorems for projective and weakly projective Boolean algebras, and (iii) indicate the overall proof strategy (reduction to the countable case via a suitable chain condition and verification of the property on free products). This will allow readers to assess the claims without immediately consulting the body of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract states the paper's aim as providing a characterization via a modification of the Freese-Nation property, but supplies no equations, definitions of the modification, theorems, or proofs. No load-bearing derivation steps are visible that could reduce by construction to inputs, self-citations, or fitted parameters. Without specific text exhibiting a self-definitional loop or renamed prediction, the derivation cannot be shown to be circular; the characterization stands as an independent claim pending full manuscript inspection.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5533 in / 901 out tokens · 16440 ms · 2026-06-27T14:11:18.498224+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

17 extracted references

  1. [1]

    Bandlow , Absolutes of compact spaces with minimal acting group, Proceedings of American Math

    I. Bandlow , Absolutes of compact spaces with minimal acting group, Proceedings of American Math. Society, 122:261--264, 1994

    1994

  2. [2]

    Balcar, T

    B. Balcar, T. Jech and J. Zapletal , Semi-Cohen Boolean algebras, Ann. of Pure and Appl. Logic 87,3 (1997), 187--208

    1997

  3. [3]

    B a k and A

    J. B a k and A. Kucharski , Topological spaces with the Freese--Nation property , Ann. Math. Sil. 33 (2019), no. 1, 41--54

    2019

  4. [4]

    B aszczyk, A

    A. B aszczyk, A. Kucharski, S. Turek , Boolean algebras admitting a countable minimally acting group . Centr. Eur. J. Math. 12(1), 46--56. (2014)

    2014

  5. [5]

    Freese, J

    R. Freese, J. B. Nation , Projective lattices , Pacific Journal of Mathematics, 75 (1975), 93--106

    1975

  6. [6]

    Daniels, K

    P. Daniels, K. Kunen and H. Zhou On the open-open game,

  7. [7]

    Halmos , Injective and projective Boolean algebras, Lattice Theory, Proceedings of Symposia in Pure Math., 2 Amer

    P. Halmos , Injective and projective Boolean algebras, Lattice Theory, Proceedings of Symposia in Pure Math., 2 Amer. Math. Soc, Providence, RI, 1961

    1961

  8. [8]

    Heindorf, L

    L. Heindorf, L. B. Shapiro Nearly Projective Boolean Algebras, Lecture Note in Mathematics 1596, Springer-Verlag 1994

    1994

  9. [9]

    Haydon , On a problem of Pelczynski: Miljutin spaces, Dugundji spaces and AE (0) -dim

    R. Haydon , On a problem of Pelczynski: Miljutin spaces, Dugundji spaces and AE (0) -dim. Studia Mathematica, 52(1974), 23--31

    1974

  10. [10]

    Koppelberg Characterizations of Cohen algebras ; in: Papers on General Topology and Applications; ed

    S. Koppelberg Characterizations of Cohen algebras ; in: Papers on General Topology and Applications; ed. S. Andima, R. Kopperman, P. R. Misra, M. E. Rudin, A. R. Todd; Annals of the New York Academy of Sciences, vol. 704 (1993), 222--237

    1993

  11. [11]

    Koppelberg , General theory of Boolean algebras ; vol 1, Handbook of Boolean Algebras; ed

    S. Koppelberg , General theory of Boolean algebras ; vol 1, Handbook of Boolean Algebras; ed. J. D. Monk and R. Bonnet, North-Holland 1989. Amsterdam 1989

    1989

  12. [12]

    Koppelberg , Projective Boolean algebras ; Chapter 20, 741--774, in vol

    S. Koppelberg , Projective Boolean algebras ; Chapter 20, 741--774, in vol. 3 of: J. D. Monk with R. Bonnet (Eds.), Handbook of Boolean algebras, North-Holland, Amsterdam etc. 1989

    1989

  13. [13]

    Koppelberg, S

    S. Koppelberg, S. Shelah , Subalgebras of Cohen algebras need not be Cohen. Logic: from foundations to applications (Staffordshire, 1993), 261--275, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996

    1993

  14. [14]

    L. B. Shapiro , Spaces that are co-absolute to a

  15. [15]

    L. B. Shapiro , On spaces that are coabsolute with dyadic compacta\/ , (Russian) Dokl. Akad. Nauk SSSR 293,5 (1987), 1077--1081

    1987

  16. [16]

    E. V. Shchepin , Functors and uncountable powers of compacta, (Russian)Uspekhi Mat. Nauk 36 (1981), no. 3(219), 3 - 62

    1981

  17. [17]

    Valov , I-favorable spaces: revisited, Topol

    V. Valov , I-favorable spaces: revisited, Topol. Proc., 51 (2018), pp. 277--292

    2018