Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

Andrzej Kucharski; S{\l}awomir Turek; Tomasz Kania

REVIEW 2 major objections 5 minor 17 references

Boolean algebras are projective exactly when they carry a coherent finite-separation map on a meet-closed base.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 12:00 UTC pith:SAGBWPYK

load-bearing objection The (FNS)* characterization of projective and weakly projective Boolean algebras is new and likely right, but the posted arXiv record has an abstract/body mismatch and a load-bearing black box in the limit step. the 2 major comments →

arxiv 2606.09580 v2 pith:SAGBWPYK submitted 2026-06-08 math.LO

Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

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

classification math.LO MSC 06E0506E10

keywords projective Boolean algebrasweakly projectiveFreese–Nation property(FNS)* propertymeet-closed baseCohen algebrasrelatively complete subalgebrasjump sets

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Projective Boolean algebras are exactly those that admit a meet-closed base carrying a coherent finite-separation map — a strengthening of the Freese–Nation property that makes separating witnesses compatible under meets. The same condition on a π-base characterizes weak projectivity. The paper builds such maps stage by stage along continuous chains of relatively complete subalgebras, with the key coherence rule preventing the witnesses from drifting apart as the chain grows. A global version of the condition holds only for countable algebras, so the characterizations are inherently local.

Core claim

The paper introduces the (FNS)∗ property for a meet-closed subset X of a Boolean algebra: a finite set map s separates disjoint elements, and the coherence rule s(a∧b)⊆(s(a)∪s(b))^∧ holds. Theorem 2 asserts that a Boolean algebra is projective if and only if it has a base X closed under meets with (FNS)∗, and Theorem 3 gives the analogous statement for weak projectivity with a π-base. The proof of existence (Theorem 5) starts from a continuous chain of relatively complete subalgebras and defines jump sets d(b) recording strict drops of the projection; the separating map at each stage is assembled from the jump-set data, with two imported lemmas guaranteeing finiteness and the existence of co

What carries the argument

The central object is the (FNS)∗ property on a meet-closed base: a finite separating map s together with coherence under meets, so that the separating witness for a meet is built from witnesses of the factors. The construction machinery is the jump-set method along a continuous chain of relatively complete subalgebras — the jump set d(b) collects the ordinal stages where the canonical projection q_α(b) strictly decreases, and the finite separation map at a limit stage is the union over these finitely many jumps, with the coherence rule ensuring that meets of base elements inherit the separation data.

Load-bearing premise

The transfinite construction works only if the imported finiteness lemmas hold — jump sets d(b) must be finite and disjoint elements must share a jump stage with disjoint projections — and if either lemma fails for arbitrary continuous chains of relatively complete subalgebras, the limit-step map becomes infinite or fails to separate.

What would settle it

Run the Theorem 5 construction on a continuous chain of relatively complete subalgebras in which some jump set d(b) is infinite, or in which two disjoint base elements never have a common jump stage with disjoint projections; if the assembled map at the first limit ordinal is infinite or fails to separate, the characterization collapses. Equivalently, exhibit a non-projective Boolean algebra with a meet-closed base satisfying (FNS)∗.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

A Boolean algebra is projective precisely when it has a meet-closed base with (FNS)∗, replacing the diagrammatic lifting definition with a concrete combinatorial witness.
Weak projectivity is captured the same way using a π-base, and Corollary 2 identifies Cohen algebras as uniform-density algebras whose π-base carries (FNS)∗.
For algebras of size at most ω1, the ordinary (FNS) property already yields (FNS)∗ on a suitable base, recovering the known coincidence between projective and rc-filtered algebras in that cardinality range.
No uncountable Boolean algebra can have a coherent finite-separation map on its whole positive cone; the projective characterization is essentially local.

Where Pith is reading between the lines

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

The coherence rule (∗) is probably the minimal compatibility condition that turns a pointwise separation property into a structural one; a natural test is whether relaxations such as containment up to joins still yield projectivity.
The jump-set method suggests that the (FNS)∗ base can be read as a 'separating skeleton' of a projective algebra; comparing minimal sizes of such bases with the algebra's π-weight might yield new cardinal invariants.
The countable-case global result hints that the boundary between local and global finite-separation behaviour coincides with the boundary at which projectivity and rc-filteredness diverge (uncountable), so the new property may sharpen the known counterexample separating those classes.
For free Boolean algebras, the construction gives an explicit separating map; one could compute the minimal finite separating sets for finite meets of atoms to check whether the coherence rule forces extra elements beyond a natural minimal choice.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The (FNS)* characterization of projective and weakly projective Boolean algebras is new and likely right, but the posted arXiv record has an abstract/body mismatch and a load-bearing black box in the limit step.

read the letter

First thing to know: the actual contribution is Theorems 2 and 3 — projectivity and weak projectivity characterized by existence of a meet-closed base (or pi-base) carrying (FNS)*, the coherent finite-separation property. That is genuinely new. Heindorf–Shapiro showed plain (FN) misses projectivity; (FNS)* with the coherence rule s(a∧b) ⊆ (s(a)∪s(b))∧ is a sensible repair, and the proofs use it exactly where needed: Theorem 6 glues countable subalgebras, Theorem 11 handles the π-base case. The constructions in Theorem 5 (jump sets, limit-step formula) are in the right shape, and the characterizations land in external theorems (Shchepin/Haydon/Koppelberg; Shapiro/Balcar–Jech–Zapletal), so no circularity. If the limit-step lemmas are as cited, the mathematics is sound. I checked enough of the successor step and the coherence verification to believe the core is right.

Now the soft spots, roughly in order. Most serious: the arXiv abstract advertises a global characterization (positive cone carries a coherent FNS map iff the algebra is countable) that the body neither proves nor even states — the body lists it as Question 1. Either the abstract is from a later version, or the paper is missing a theorem. As posted, the claim set is ambiguous. The metadata also lists T. Kania as an author while the manuscript's title page has only Kucharski and Turek. That needs fixing before anyone cites it as-is.

Second: the limit step in Theorem 5 imports two lemmas from Heindorf–Shapiro without statement or proof. The stress-test worry is specifically [6, Lemma 2.1.3] and whether it really gives β ∈ d(a) ∩ d(b), not just union. If the lemma only gives union, the separating witnesses don't land in the intersection and the construction collapses at the first limit ordinal. This is load-bearing. The authors should restate the lemmas, or better, prove them in the context of Koppelberg chains. If they can't, the proof has a real hole.

Third, minor: in Theorem 5's successor step there's a "without loss of generality" over the two cases that is repairable by minimality of q; and in Theorem 8 the text says regular where relatively complete is needed, a slip that doesn't affect the argument. These are small.

Bottom line: the core idea and the two main theorems are worth a serious referee. It's not a desk reject. But the posted record as-is isn't something I'd cite without checking the black-box lemmas and the abstract/body mismatch. I'd send it to review with a request for those fixes.

Referee Report

2 major / 5 minor

Summary. The paper introduces a coherent strengthening of the Freese--Nation separation property, denoted (FNS)*, defined on meet-closed subsets of a Boolean algebra. The additional coherence condition requires s(a∧b) ⊆ (s(a)∪s(b))^∧. The main claims are Theorem 2: a Boolean algebra is projective iff it has a meet-closed base carrying an (FNS)*-map, and Theorem 3: it is weakly projective iff it has a meet-closed π-base carrying such a map. The 'if' directions use the existence of a club of countable relatively complete/regular subalgebras and then invoke the known structural characterizations of Shchepin/Haydon/Koppelberg and Shapiro/Balcar--Jech--Zapletal/Bandlow. The 'only if' directions are proved by transfinite construction along a Koppelberg chain. The paper also proves the global version is much stronger: a whole-algebra (FNS)* map exists only in the countable case, so the projective/weakly projective characterizations are intrinsically local.

Significance. If the construction is correct, this is a substantial contribution. The paper identifies a genuinely local, finite-separation condition that repairs the failure, established by Heindorf--Shapiro, of plain (FNS) or (FN) to characterise projectivity. The coherence rule (∗) is the key new idea; it is simple, parameter-free, and makes the separations compatible with subalgebra chains. The weakly projective analogue via π-bases and the connection to Cohen algebras are natural and potentially useful. The paper is also honest about its limitations, explicitly leaving open whether every projective algebra has (FNS)* on the whole algebra. A particular strength is that the two directions land in well-established external characterizations, so the burden on the new notion is clear and falsifiable.

major comments (2)

[§3, Theorem 5 (successor step)] In the verification that s_{α+1} has the (FNS) property, the proof assumes q_α^{α+1}(a)∧q_α^{α+1}(b)>0 and then says 'Without loss of generality, we can assume that a,b∈{x∧y∧(−1)^i e_α: ...}'. This is not justified: X_{α+1} contains X_α as well as the new elements, and the displayed contradiction relies on both elements having the form x∧y∧(−1)^i e_α. The mixed case, where one of a,b lies in X_α, is not handled. In that case the claimed dichotomy -- either the q-images are disjoint or the two elements lie on opposite sides of e_α -- is not proved. A minimality-of-q argument can probably repair this, but as written this is a gap in the main construction and it propagates to both Theorem 2 and Theorem 3.
[§3, Theorem 5 (limit step)] The definition of s_α and the verification of (FNS) at limit ordinals rest entirely on two imported lemmas from Heindorf--Shapiro: [6, Lemma 2.1.1] (finiteness of the jump set d(a)) and [6, Lemma 2.1.3] (existence of β∈d(a)∩d(b) with q_{β+1}^α(a)∧q_{β+1}^α(b)=0, or the analogous statement for q_0^α). These lemmas are load-bearing: they ensure that s_α(a) is finite and that the separating pair lies in s_α(a)∩s_α(b). The paper neither states these lemmas nor verifies that the Koppelberg chain obtained from Theorem 4(6) satisfies their hypotheses. If Lemma 2.1.3 only guarantees β∈d(a)∪d(b), or if the hypotheses differ, the limit-step construction collapses at the first limit ordinal. Please state the lemmas and check the hypotheses explicitly.

minor comments (5)

[§2, Definition of base/π-base] The definition is garbled by the parenthetical 'each element of A+ (X is a dense subset of A)'. The base case and π-base case should be stated separately.
[§3, Theorem 6 proof] The symbol D is used both for a countable subset of A and for the family of witnesses. Use different notation, e.g. ℕD for the countable family, to avoid confusion.
[§3, Theorem 8 proof] The proof says 'By Lemma 4 we have A_A ≤ reg A' and then claims the subalgebras are relatively complete. To get relative completeness one must apply the 'moreover' part of Lemma 4, noting that P=A_A\{0} is a base of A_A and X=A is a base of A. This should be said explicitly.
[§4, Theorem 11 proof] The proof begins 'Let B be a dense subset of A', but the hypothesis gives a π-base X; the letter B is also used for the whole algebra. This obscures the application of Lemma 3 and should be cleaned up.
[Throughout] There are several typographical errors: 'proprety', 'is ismorphic', the malformed expression 'β, * β' in clause (7), and the numbering 'Theorem 2' in the proof header after Theorem 11. These should be corrected.

Circularity Check

0 steps flagged

No significant circularity: (FNS)* is a fresh property and both directions of Theorems 2-3 are anchored in external characterizations.

full rationale

The paper's central derivations are not circular. (FNS)* is introduced as a new, independent strengthening of (FNS) rather than being defined in terms of projectivity or weak projectivity. The 'if' directions of Theorems 2 and 3 convert (FNS)* into an external characterization: Theorem 6 produces a countable rc-tower via Lemma 4, then applies Theorem 4 (Shchepin/Haydon/Koppelberg); Theorem 11 produces a club of countable regular subalgebras and applies Theorem 9 (Shapiro/Balcar–Jech–Zapletal/Bandlow). The 'only if' directions use external characterizations (Theorem 4(6) for projective, Theorem 9(2) for weakly projective, via a dense projective subalgebra). The limit-step verification in Theorem 5 relies on Heindorf–Shapiro's monograph [6, Lemma 2.1.1 and Lemma 2.1.3]; these are external prior results, not self-citations, and they are not equivalent to the target theorem. Even if those lemmas were insufficient in the asserted generality, that would be a correctness gap, not circularity. The only self-citation, [3], appears in Lemmas 1 and 3 alongside the external references [9] and [10] for elementary facts about the q-map and regular subalgebras; it is not load-bearing for the main equivalence. The paper also explicitly leaves open whether (FNS)* extends to the whole projective algebra (Question 1), confirming that the property is not being identified with the known characterization by fiat.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

Pure mathematics: no fitted parameters. The load-bearing inputs are: (i) the standard external characterizations of projectivity (Theorem 4: Shchepin 1981, Haydon 1974, Koppelberg 1989) and weak projectivity (Theorem 9: Shapiro 1987, Balcar–Jech–Zapletal 1997, Bandlow 1994, Koppelberg) — the paper's iff directions terminate in these; (ii) two jump-set lemmas from Heindorf–Shapiro's monograph ([6, Lem. 2.1.1] finiteness of d(a); [6, Lem. 2.1.3] common-jump witness for disjointness) that power the limit step of the main construction but are neither proved nor restated; (iii) elementary q-map/regularity facts (Koppelberg [9]; [3]). The only new object is (FNS)* itself, which is the contribution rather than a borrowed assumption.

axioms (5)

domain assumption Heindorf–Shapiro [6, Lem. 2.1.1]: jump sets d(a) are finite for chains with A_{α+1} = A_α(e_α)
Used to ensure s_α(a) is finite in the limit step of Theorem 5; not proved in the paper.
domain assumption Heindorf–Shapiro [6, Lem. 2.1.3]: disjointness of a,b ∈ X_α is witnessed at a common jump ordinal
The (FNS) verification at limit stages (Theorem 5) depends on finding β ∈ d(a)∩d(b) with the stated separation; imported, not proved.
domain assumption Theorem 4(6) (Koppelberg): projective BAs are unions of continuous chains with countable successor steps A_{α+1} = A_α(e_α), A_α ≤rc A_{α+1}
The entire construction of X and s in Theorem 5 runs along this chain; projectivity is used exactly through this external characterization.
domain assumption Theorem 9 (Shapiro/Balcar–Jech–Zapletal/Bandlow/Koppelberg): weak projectivity = dense projective subalgebra = club of countable regular subalgebras closed under pairwise union
Both directions of Theorem 3 terminate in these external characterizations.
standard math Lemma 3 equivalences (regular subalgebras) and Lemma 1 properties of the q-map (Koppelberg [9]; also [3])
Elementary background used in Lemma 4 and Theorem 11; standard in the field.

invented entities (1)

(FNS)*-map s on a meet-closed base X no independent evidence
purpose: A finite-witness separation map with the coherence rule s(a∧b) ⊆ (s(a)∪s(b))^∧; the paper's new certificate for projectivity and weak projectivity.
A defined mathematical structure, not a physical postulate. Its only handles are the two characterization theorems themselves; the paper does not test the criterion on the known Heindorf–Shapiro rc-filtered non-projective algebra, which would be the natural falsifiable check.

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Cite this review

Pith. "Pith review of Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations." pith.science (2026). https://pith.science/paper/SAGBWPYK

@misc{pith2026260609580,
  author       = {Pith},
  title        = {Pith review of: Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SAGBWPYK}},
  note         = {Machine review of arXiv:2606.09580}
}

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read the original abstract

We isolate a coherent finite-separation strengthening of the Freese--Nation property and use it to characterise projective and weakly projective Boolean algebras. A Boolean algebra is projective if and only if it has a meet-closed decomposition base carrying such a coherent finite-separation map; it is weakly projective if and only if it has a meet-closed $\pi$-base carrying one. The global form of the same property is much stronger: the positive cone of a Boolean algebra carries a coherent finite-separation map if and only if the algebra is countable. Thus the projective and weakly projective characterisations are intrinsically local and cannot be strengthened by requiring the whole algebra to carry the coherent map except in the countable case.

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

17 extracted references

[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

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[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]

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]

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]

Freese, J

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

1975
[6]

Daniels, K

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

Heindorf, L

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

1994
[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]

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]

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]

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]

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]

L. B. Shapiro , Spaces that are co-absolute to a
[15]

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

1987
[16]

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

1981
[17]

Valov , I-favorable spaces: revisited, Topol

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

2018

[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

Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it