Decompositions of set-valued mappings

Igor Protasov

REVIEW 4 major objections 5 minor 6 references

Decompositions of set-valued mappings

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any set-valued map with bounded fibres and bounded preimages, fewer than $\kappa$ bijections reconstruct every fibre.

desk verdict The main theorem is false; a two-element counterexample satisfies every hypothesis but violates the conclusion, so the applications in Section 2 cannot stand as written. read the letter →

arxiv 1908.03911 v2 pith:CUQJ3JTI submitted 2019-08-11 math.GN

Igor Protasov This is my paper · ORCID

classification math.GN MSC 03E0554E05

keywords set-valuedmappingselectorbijectiveballeancoarsespaceG-spacegroupidealcardinalinvariant

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

This paper proves a decomposition theorem for set-valued mappings. If $F:X\to B_X$ satisfies $x\in F(x)$ and both the sets $F(x)$ and the preimages $F^{-1}(x)$ have size below some infinite cardinal $\kappa$, then all the values $F(x)$ can be produced pointwise by fewer than $\kappa$ bijective selectors of $X$. The theorem matters because it turns a pure cardinal-size condition into concrete permutation structure: every fibre is the image of $x$ under a small family of global bijections. The paper then applies this to coarse geometry, showing that balleans, or coarse spaces, whose balls are uniformly smaller than $\kappa$ admit $G$-space representations in which the defining group ideal also has all members smaller than $\kappa$, and that finitary cellular balleans come from locally finite permutation groups.

What carries the argument

The load-bearing object is the intersection graph $\Gamma$ whose vertices are the values $F(x)$ and whose edges join intersecting values. The proof bounds the local degree of $\Gamma$: if $m$ exceeds all $|F(x)|$ and all $|F^{-1}(x)|$, then no vertex meets more than $m^2-1$ other vertices, so the graph is $m^2$-colourable. Each colour class is a family of pairwise disjoint fibres, and on each class the proof defines a block of global transpositions by pairing the anchor point of each fibre with each listed element. Combining these blocks for all colour classes and all index positions $j<M$ yields the family $\mathcal{F}$ whose pointwise images reproduce every fibre. In the uncountable case, the same block construction runs on small pieces obtained by closing points under iterated applications of $F$ and $F^{-1}$.

What would settle it

Enumerate all set-valued maps on a finite set, say $\{1,2,3,4\}$, that satisfy $x\in F(x)$, and compute for each the least number of bijections needed to realize every fibre as $\{f(x):f\in\mathcal{F}\}$. If any instance satisfying the cardinal bounds requires more than $\max_x |F(x)|$ bijections, the claimed bound is false; if none does, the finite case of the theorem is confirmed.

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Extended reading notes

Core claim

The central claim is Theorem 1: for any set $X$ and any set-valued mapping $F:X\to B_X$ with $x\in F(x)$, $\sup_x |F(x)|<\kappa$, and $\sup_x |F^{-1}(x)|<\kappa$, there is a family $\mathcal{F}$ of bijective selectors with $|\mathcal{F}|<\kappa$ such that $F(x)=\{f(x):f\in\mathcal{F}\}$ for every $x$. In the case $\kappa=\omega$, the proof forms the intersection graph of the family $\{F(x)\}$, colours it so that intersecting fibres receive different colours, and then defines, for each colour class and each $j$ below a common size bound, a bijection that transposes the anchor point of each fibre with its $j$-th element. The uncountable case is reduced to this construction by partitioning $X$ into pieces on which the map is small. Theorem 2 applies the decomposition to balleans: if every ball has size below $\kappa$, the ballean is asymorphic to a ballean $(X,G,\mathcal{I})$ generated by a group of permutations and a group ideal whose members all have size below $\kappa$. Theorem 3 specialises this to finitary cellular balleans and locally finite permutation groups.

Load-bearing premise

The proof's construction depends on treating each $F(x)$ as a private container: swapping two points inside $F(y)$ must not push any element outside its own $F$-set, and the stated cardinal bounds do not by themselves guarantee that container property.

Editorial extensions

If this is right

Any set-valued map with $x\in F(x)$ and all fibres and preimages of size below $\kappa$ can be presented pointwise by fewer than $\kappa$ permutations, not just by arbitrary selectors.
Every ballean whose balls are uniformly of size below $\kappa$ is asymorphic to a $G$-space ballean in which the group ideal also has all members of size below $\kappa$.
The $\kappa=\omega$ case recovers the earlier representation of finitary balleans by group ideals made of finite sets.
Finitary cellular balleans are, up to asymorphism, exactly the finitary balleans of locally finite permutation groups.

Reading between the lines

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

The swap construction reads $F$ as an undirected relation: for each transposition to stay inside $F(x)$, one needs $y\in F(x)$ whenever $x\in F(y)$. The two cardinal bounds alone do not force this, so a natural extension is to prove the decomposition under a mutual-membership or symmetry condition, or to exhibit an asymmetric $F$ that resists the construction.
The number of bijections needed to cover all fibres is bounded by the chromatic number of the intersection graph; comparing that number with the maximum fibre size could give a sharper invariant for set-valued maps.
For balleans, the theorem suggests a dictionary between cardinal bounds on balls and cardinal bounds on group ideals in $G$-space representations; a testable extension is whether the cellular hypothesis in Theorem 3 can be relaxed while keeping the permutation group locally finite.

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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

Punchline: the central theorem of this short note is false, and the counterexample is small enough to check on the back of an envelope. Take X = {a, b}, F(a) = {a, b}, F(b) = {b}. Then x ∈ F(x) for both x, and both sup_x |F(x)| and sup_x |F^{-1}(x)| equal 2 < ω, so the hypotheses of Theorem 1 hold at κ = ω. But every bijective selector must be the identity: b can only map to b, so a must map to a. The set {f(a) : f is a bijective selector} is {a}, not {a, b}. The conclusion fails.

The proof breaks at the transposition construction. For each j, the chosen map swaps y_α and y_αj on every F(y_α) in a color class. If y_α is not in F(y_αj), the swap moves y_αj outside its allowed target set. The hypotheses bound cardinalities but say nothing about overlap between the sets F(x), so nothing prevents exactly this situation. The idea of extending the κ = ω case from [4] to arbitrary infinite cardinals is natural, but the theorem as stated is not a valid extension.

What the paper does well: it is concise, clearly motivated by ballean/G-space questions, and the proof strategy is transparent. The notation is workable, and the references to [3] and [4] look appropriate; I do not see circular reasoning or a citation-pattern problem.

Because Theorems 2 and 3 both invoke Theorem 1, they inherit the failure. I am not saying the applications are unrecoverable: a corrected theorem with an additional hypothesis, perhaps that the family {F(x)} forms a partition or that the overlaps are controlled, might restore some of the claims. But that is future work, not what the manuscript currently proves.

Recommendation: I would not send the current version to a referee. I would communicate the counterexample to the author and invite a corrected statement if one exists. The paper is short, and a clean corrected version would be more useful than a refereeing cycle on a false theorem.

Referee Report

4 major / 5 minor

Summary. The paper claims a decomposition theorem for set-valued mappings: under the hypotheses that F(x) contains x, and that sup_x |F(x)| and sup_x |F^{-1}(x)| are both bounded below an infinite cardinal κ, there exists a family of bijective selectors of size < κ such that each F(x) equals the set of images of x under these selectors. The paper applies this theorem to construct G-space representations of balleans with ideals of size < κ, and also states a separate result for finitary cellular balleans. The central proof constructs selectors as transpositions on color classes of an intersection graph on the family of F(x), and the applications rely directly on this construction.

Significance. If the decomposition theorem were correct, it would provide a clean cardinality control on selector families and would strengthen prior results on balleans of bounded geometry, with potential applications to coarse geometry and G-space representations. The paper is short, self-contained, and the intended proof strategy is natural. However, the central claim is false: a two-element example satisfies all hypotheses but violates the conclusion. Since both the main theorem and the ballean application in Theorem 2 depend on this claim, the contribution as written cannot stand. The paper also includes a statement of Theorem 3 for finitary cellular balleans, but that proof is independent of the flawed theorem and may be salvageable, though it is not enough to rescue the paper's main claim.

major comments (4)

[Section 1, Theorem 1, Case κ = ω] The proof of Theorem 1 is invalid because the global transposition defined for each color class does not necessarily preserve the selector condition. The proof states that for each j < M, the function f_j acts as a transposition of y_α and y_{α_j} on each F(y_α) in a color class and identically at all other elements. But if an element x = y_{α_j} belongs to some F(y_β) in a different color class, then the transposition may map x to y_α, which need not lie in F(x). The hypotheses only bound cardinalities of F(x) and F^{-1}(x); they do not imply the required disjointness or the containment y_α ∈ F(y_{α_j}).
[Section 1, Theorem 1, Case κ = ω] Theorem 1 is false as stated. Let X = {a,b} and define F(a) = {a,b}, F(b) = {b}. Then x ∈ F(x) for each x, sup_x |F(x)| = 2 < ω, and sup_x |F^{-1}(x)| = 2 < ω, so the hypotheses hold with κ = ω. The only bijective selector is the identity map, since any bijection must send b to a value in F(b) = {b}. Hence {f(a) : f is a bijective selector} = {a}, which is not equal to F(a) = {a,b}. Thus the conclusion of Theorem 1 fails.
[Section 2, Theorem 2] Theorem 2 applies Theorem 1 to the ball mappings F_E(x) = E[x]. Since Theorem 1 is false, the proof of Theorem 2 is not established. Moreover, the same counterexample can be embedded as a ballean on a two-point set to show that the conclusion of Theorem 2, as stated with arbitrary κ, would require an independent argument rather than the present appeal to Theorem 1.
[Section 1, Case κ > ω] The case κ > ω proceeds by partitioning X into blocks P that are closed under F and F^{-1}, then invoking the case κ = ω on each block. This relies on the truth of the κ = ω case, which is false. Additionally, the proof asserts without demonstration that the case κ = ω can be applied within each block to obtain a family of bijective selectors of the whole mapping F, but the external construction from the flawed case does not supply such selectors.

minor comments (5)

[Title and abstract] The title contains a typographical artifact: 'SET-V ALUED' should be 'SET-VALUED'.
[Section 1, proof of Theorem 1] The phrase 'bijective selectors of X' in the statement of Theorem 1 should be 'bijective selectors of F', since the selectors are functions into X but the object being selected is the mapping F.
[Section 1, proof of Theorem 1] The notation F^{-1}F(y) is used without definition; it presumably means {x ∈ X : F(x) ∩ F(y) ≠ ∅}, but this should be stated explicitly.
[Section 1, proof of Theorem 1] The enumeration 'F(y_α) = {y_{α_j} : j < M}' with repetitions is unclear, and the subsequent definition of f_j as 'a transposition of y_α and y_{α_j} at each F(y_α)' is ambiguous when the same pair recurs or when y_α = y_{α_j}. A precise definition of the map on all of X is needed.
[Section 1, proof of Theorem 1] The reference '[2]' is attributed to 'Harary, Graph Theory' but the standard citation is 'Harary, Graph Theory, Addison-Wesley, 1969'; the listed 1994 edition is acceptable if that is the source used, but the author name is misspelled as 'A. Harary'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main decomposition is a direct combinatorial construction, and the cited prior G-space theorem is external background, not an input that is equivalent to the conclusion.

full rationale

The derivation of Theorem 1 is a direct construction from graph coloring: the family of bijective selectors is built by transpositions on color classes of the intersection graph Γ, using only the standard degree-chromatic number bound from [2] and elementary cardinal arithmetic. No fitted parameter is renamed as a prediction; F is not defined in terms of the selectors, and the selectors are constructed after F is given. The case κ > ω reuses the argument pattern of the ω case but does not assume the theorem's conclusion: it first forms σ-small closed pieces and then applies the same transposition construction on the enumerated pieces. Citations [3] and [4] are prior results in ballean theory; in particular, the G-space representation theorem from [3] is used as background context for the application, not as the proof of Theorem 1. Even in Theorem 2, the cardinal bound |A| < κ is derived from Theorem 1 rather than from [3], so the earlier self-citation is not load-bearing in a way that reduces the new claim to its own input. The reader-identified counterexample concerns a missing disjointness/selection hypothesis in the transposition argument; that is a correctness flaw, not circularity, and under the given instructions it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No fitted parameters or invented entities. The only non-external input beyond standard graph theory is the false implicit disjointness and selection assumption, listed as an axiom because the proof depends on it and it is not granted by the hypotheses.

assumptions (2)

standard math If the local degree of each vertex of a graph is at most k, then the chromatic number is at most k+1.
Used to color the family of sets F(x) in the finite case of Theorem 1; this is a standard graph theory fact cited as [2] and is not the source of the error.
ad hoc to paper Each element x of X is affected by transpositions from at most one color class of the partition of {F(x)}, so the global transpositions remain selectors for every x.
Implicitly assumed in the proof of Theorem 1 when f_j is defined as a transposition of y_alpha and y_alpha_j on every F(y_alpha) in a color class. The theorem's hypotheses do not imply this; in the counterexample, a lies in both F(a) = {a} and F(b) = {a,b}.

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Cite this review

Pith. "Pith review of Decompositions of set-valued mappings." pith.science (2026). https://pith.science/paper/CUQJ3JTI

@misc{pith2026190803911,
  author       = {Pith},
  title        = {Pith review of: Decompositions of set-valued mappings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CUQJ3JTI}},
  note         = {Machine review of arXiv:1908.03911}
}

abstract

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \longrightarrow B_{X}$ be a set-valued mapping such that $x \in F(x)$, $sup_{x\in X} | F(x)|< \kappa$, $sup_{x\in X} | F^{-1}(x)|< \kappa$ for all $x\in X$ and some infinite cardinal $\kappa$. Then there exists a family $\mathcal{F}$ of bijective selectors of $F$ such that $|\mathcal{F}|<\kappa$ and $F(x) = \{ f(x): f\in\mathcal{F}\}$ for each $x\in X$. We apply this result to $G$-space representations of balleans.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

[4]

Protasov, Balleans of bounded geometry and G-space, Algebra Discrete Math

I.V. Protasov, Balleans of bounded geometry and G-space, Algebra Discrete Math. 2008, no 2, 101-108

work page 2008
[3]

O. V. Petrenko, I.V. Protasov, Balleans and G-spaces, Ukr. Mat. Zh. 64 (2012), 344-350

work page 2012
[1]

On the space of ends of infinitely generated groups

Y. Cornulier, On the space of ends of inﬁnitely generated groups, arXiv: 1901.11073

work page Pith review arXiv 1901
[2]

Harary, Graph Theory, Addison-Wesley, 1994

A. Harary, Graph Theory, Addison-Wesley, 1994

work page 1994
[5]

Protasov, M

I. Protasov, M. Zarichnyi, General Asymptology, Mat. Stud. Monogr. Ser, vol. 12, VNTL, Lviv, 2007

work page 2007
[6]

Roe, Lectures on Coarse Geometry , Univ

J. Roe, Lectures on Coarse Geometry , Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003. CONTACT INFORMATION I. Protasov: Faculty of Computer Science and Cybernetics Kyiv University Academic Glushkov pr. 4d 03680 Kyiv, Ukraine i.v.protasov@gmail.com

work page 2003

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [4]

Protasov, Balleans of bounded geometry and G-space, Algebra Discrete Math

I.V. Protasov, Balleans of bounded geometry and G-space, Algebra Discrete Math. 2008, no 2, 101-108

work page 2008

[2] [3]

O. V. Petrenko, I.V. Protasov, Balleans and G-spaces, Ukr. Mat. Zh. 64 (2012), 344-350

work page 2012

[3] [1]

On the space of ends of infinitely generated groups

Y. Cornulier, On the space of ends of inﬁnitely generated groups, arXiv: 1901.11073

work page Pith review arXiv 1901

[4] [2]

Harary, Graph Theory, Addison-Wesley, 1994

A. Harary, Graph Theory, Addison-Wesley, 1994

work page 1994

[5] [5]

Protasov, M

I. Protasov, M. Zarichnyi, General Asymptology, Mat. Stud. Monogr. Ser, vol. 12, VNTL, Lviv, 2007

work page 2007

[6] [6]

Roe, Lectures on Coarse Geometry , Univ

J. Roe, Lectures on Coarse Geometry , Univ. Lecture Ser., vol. 31, American Mathematical Society, Providence RI, 2003. CONTACT INFORMATION I. Protasov: Faculty of Computer Science and Cybernetics Kyiv University Academic Glushkov pr. 4d 03680 Kyiv, Ukraine i.v.protasov@gmail.com

work page 2003