A comparison of arithmetical operations with $f$ correlated fuzzy numbers

Diogo Sampaio da Silva; Roberto Antonio Cordeiro Prata

REVIEW 4 major objections 4 minor 19 references

A comparison of arithmetical operations with $f$ correlated fuzzy numbers

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For f-correlated fuzzy numbers, the paper claims the correlated sum equals the ordinary sum level by level, and the correlated product is contained in the ordinary product.

desk verdict The paper's central sum-comparison theorem is false for decreasing f; the proof separates x and f(x) in {x+f(x)}, and the paper's own f(x)=-x case contradicts it. read the letter →

arxiv 2412.00046 v3 pith:R7J3YDML submitted 2024-11-23 math.GM

Diogo Sampaio da Silva , Roberto Antonio Cordeiro Prata This is my paper · ORCID

classification math.GM MSC 03E7226E50

keywords fuzzynumbersf-correlatedinteractivearithmeticaloperationsalpha-levelsextensionprincipleintervalarithmeticcorrelatedinverses

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 is about a special class of interactive fuzzy numbers: pairs $(A,B)$ in which $B$ is obtained from $A$ by a continuous monotone injective function $f$, so the joint uncertainty of the pair is concentrated on the curve $y=f(x)$. The authors derive direct formulas for the correlated sum and product of such pairs using only real operations, interval arithmetic, and the function $f$. Their main theorem says that the correlated sum coincides with the standard fuzzy sum for every alpha-level, and that every alpha-level of the correlated product is contained in the corresponding level of the standard product. If that is right, dependent fuzzy arithmetic can be performed without constructing a full joint distribution. The paper closes by exhibiting additive and multiplicative inverses for the correlated operations.

What carries the argument

The central object is the $f$-correlated fuzzy number: two fuzzy numbers whose joint possibility distribution $\phi_C(x,y)=\phi_A(x)\chi_{\{y=f(x)\}}$ is concentrated on the graph of a continuous monotone injective $f$, so that knowing $A$ determines $B$. The carrying identities are Lemma 1, $[B]_\alpha=f([A]_\alpha)$, and Lemma 2, $[A+_f B]_\alpha=\overline{\{x+f(x):\phi_A(x)>\alpha\}}$ and $[A\cdot_f B]_\alpha=\overline{\{xf(x):\phi_A(x)>\alpha\}}$. These identities reduce correlated operations to one-dimensional sets, and Theorem 1 compares them with standard interval arithmetic by splitting infimums and supremums over sums and by embedding the diagonal set $\{xf(x)\}$ into the product set $\{x_1f(x_2)\}$.

What would settle it

Let $A$ be any fuzzy number with support $[1,2]$ and take $f(x)=-x$; at $\alpha=0$ the correlated sum is $\{0\}$ while the standard sum is $[1,2]+[-2,-1]=[-1,1]$, so this calculation contradicts the equality asserted in Theorem 1.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: for $f$-correlated fuzzy numbers $A$ and $B$ with $B=f(A)$, for every $\alpha\in[0,1]$, $[A+_f B]_\alpha=[A]_\alpha+f([A]_\alpha)$, which the authors identify with the standard sum $[A+B]_\alpha$; and $[A\cdot_f B]_\alpha\subseteq[A]_\alpha f([A]_\alpha)=[A\cdot B]_\alpha$. The correlated product level is derived as the closure of $\{xf(x):\phi_A(x)>\alpha\}$, and the proof of containment uses the fact that this diagonal set lies inside the product set $\{x_1 f(x_2):\phi_A(x_1),\phi_A(x_2)>\alpha\}$. The paper also gives explicit formulas for linear and hyperbolic $f$, and it proves that choosing $f(x)=-x$ gives an additive inverse while $g(x)=1/x$ gives a multiplicative inverse.

Load-bearing premise

The main theorem rests on assuming that the smallest and largest values of $x+f(x)$ over a level set can be obtained by adding the smallest and largest values of $x$ and $f(x)$ separately, an independence that fails when $f$ is decreasing.

Editorial extensions

If this is right

For $f(x)=qx+r$ with $q>0$, the correlated sum level is $(q+1)[A]_\alpha+r$, so it can be produced by scaling and shifting the original level.
For $f(x)=qx+r$, the correlated product level is $q[A\cdot_{\mathrm{id}} A]_\alpha+r[A]_\alpha$, so only the product of $A$ with itself and ordinary level arithmetic are needed.
For $f(x)=q/x+r$, the correlated product is $q+r[A]_\alpha$, a translated and scaled copy of $A$, and the correlated sum is $[A]_\alpha+q\overline{\{1/x:\phi_A(x)>\alpha\}}+r$.
Choosing $f(x)=-x$ gives $[A+_f B]_\alpha=\{0\}$ and choosing $g(x)=1/x$ gives $[A\cdot_g C]_\alpha=\{1\}$, so correlated operations have additive and multiplicative inverses.
The inclusion $[A\cdot_f B]_\alpha\subseteq[A\cdot B]_\alpha$ implies correlated multiplication never produces wider alpha-levels than the standard product.

Reading between the lines

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

Editorial inference: the equality proof works by separating the infimum and supremum of $x+f(x)$; framing the theorem with an explicit monotonicity hypothesis on $f$ would make the scope precise, since the separation step behaves differently for increasing and decreasing functions.
Editorial inference: the product containment suggests correlated multiplication is the diagonal restriction of ordinary fuzzy multiplication, which connects directly to constrained interval arithmetic where dependency between variables is enforced at the interval level.
Editorial inference: the inverses from Theorem 2 make it natural to define correlated subtraction via $f(x)=-x$ and correlated division via $g(x)=1/x$, opening a route to fuzzy differential equations in which the derivative and the state are interactively related.
Editorial inference: for increasing monotone dependence, the sum coincidence implies that adding two such fuzzy numbers can be done level-wise with ordinary intervals, which would simplify simulation codes for models with functional dependence between uncertain parameters.

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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

Quick take: the main theorem is false. Theorem 1 claims the f-correlated sum equals the standard sum for continuous monotone f. That equality fails for decreasing f. The proof's step from Eq. (20) to Eq. (21) replaces inf/sup of x+f(x) over the diagonal by inf x + inf f(x) and sup x + sup f(x). That treats x and f(x) as independently variable, which is only valid when the monotonicities align. For decreasing f it is generally wrong. The paper itself contains the counterexample: for f(x) = -x, Lemma 2 gives [A +_f B]_α = {0}, while Eq. (10) gives [L(α)-U(α), U(α)-L(α)]. These coincide only when A is crisp. So the central claim is internally contradicted by the authors' own formulas.

What is actually good: the product containment, [A ·_f B]_α ⊂ [A]_α · f([A]_α), is correct and follows immediately from the diagonal being a subset of the product set. The explicit formulas for linear and hyperbolic f are routine but correct, and Theorem 2 (existence of additive and multiplicative inverses via f(x)=-x and f(x)=1/x) is a nice observation that also holds.

Now the soft spots. The invalid decomposition is load-bearing; without it, Theorem 1 collapses. There is also a smaller numerical slip in Example 1: for a fuzzy number with support (-2,1), [A · id_A]_0 is {x^2 : x∈[-2,1]} = [0,4], not [0,2] as written. That indicates the example wasn't checked. The citation pattern is unobjectionable; Lemmas 1-2 come from the authors' own prior work (refs [6,7]), which is normal in a research program.

Bottom line: this isn't a paper you can build on. The correct fragments (product containment, inverse theorem, the formulas) could form a short note if the sum claim is removed or restricted to increasing f with a corrected proof. But as it stands, the paper is internally inconsistent, not just incomplete.

Recommendation: I would not send this to a referee; desk reject is appropriate. If the authors resubmit with the comparison claim fixed or dropped, it becomes a different, much smaller paper that could merit review.

Referee Report

4 major / 4 minor

Summary. The paper introduces a class of interactive fuzzy numbers, called f-correlated fuzzy numbers, where a fuzzy number B is related to a fuzzy number A through a continuous monotone injective function f. The authors derive formulas for the alpha-levels of the correlated sum and product, A +_f B and A *f B, using the diagonal set {x, f(x)}. The main claimed result (Theorem 1) is that for any continuous monotone injective f, the correlated sum of f-correlated fuzzy numbers equals the standard (non-interactive) sum, while the correlated product is a subset of the standard product. The paper also presents special cases (linear and hyperbolic f), a counterexample showing strict containment for the product, and a theorem on additive and multiplicative inverses for correlated operations.

Significance. If valid, the claim that correlated sums reduce to ordinary interval arithmetic on alpha-levels would be a convenient and powerful simplification for practitioners of interactive fuzzy arithmetic. The product-containment result is plausible and correctly argued via the inclusion of the diagonal set in the Cartesian product. However, the central sum-coincidence claim is false for decreasing f, as shown by the paper's own equations. Consequently, the main theorem does not hold in the stated generality, and the paper cannot serve as a reliable reference for computing correlated sums. The concrete example distinguishing [A *id A] from [A * A] is a useful observation, but it does not rescue the paper's principal claim.

major comments (4)

[§2.3, Eq. (20)–(21)] The proof of Theorem 1 invalidly replaces the infimum and supremum of the diagonal set {x + f(x) : φ_A(x) > α} with the separated expressions inf x + inf f(x) and sup x + sup f(x). This assumes that x and f(x) can be extremized independently, which is false for decreasing f. The paper itself provides the counterexample: with f(x) = -x, Eq. (15) (or Lemma 2) gives [A +_f B]_α = {x - x} = [0,0], whereas the standard sum computed in Eq. (10) with q = -1, r = 0 is [L(α) - U(α), U(α) - L(α)], a non-degenerate interval for any non-crisp A. Thus Theorem 1 is contradicted by the paper's own formulas and the claimed coincidence of correlated and standard sums is false for decreasing f.
[§2.3, Eq. (21)] The displayed formula in Eq. (21) also contains a typo: the rightmost term is written as 'sup xf(x)' rather than 'sup f(x)'. More substantively, the passage from Eq. (20) to Eq. (21) relies on the assertion that the infimum and supremum preserve addition, which is only an inequality (inf(x+y) ≥ inf x + inf y and sup(x+y) ≤ sup x + sup y) in general; equality requires additional monotonicity or independence assumptions that are not stated and do not hold for decreasing f.
[§2, Lemma 2] Lemma 2 defines the alpha-level sets of the correlated sum and product using the condition φ_A(x) > α, whereas the paper's introductory definition of alpha-levels (for α > 0) uses φ_A(x) ≥ α, with φ_A(x) > 0 only for α = 0. This inconsistency is not harmless: for α = 1, the strict inequality excludes the core and can change the resulting interval when the supremum is attained only at membership exactly one. The proofs of Theorems 1 and 2 inherit this issue, and the paper should clarify whether the intended definition is strict or non-strict.
[§2.1.2 and §2.2, hyperbolic case] The formulas for f(x) = q/x + r assume x ≠ 0 and do not state the domain of f relative to the support of A. If the support of A contains 0, the expressions q/L(α) and q/U(α) are undefined or infinite. This is a gap in the derivation of Eqs. (12)–(14) and (17)–(18), and it affects the claimed generality of the results for the hyperbolic case.

minor comments (4)

[Abstract] The abstract contains a typo ('fuz zy' should be 'fuzzy') and overstates the result by saying 'We proved that their correlated and standard sum coincide' when, as noted above, this is false for decreasing f.
[References] References [8] and [9] are the same work by Carlsson, Fuller, and Majlender, 'Additions of completely correlated fuzzy numbers,' with identical bibliographic details; one duplicate should be removed and citations renumbered.
[§2.3, Eq. (24)–(25)] The product-containment proof is essentially correct, but the notation is imprecise: the set {x_1 f(x_2) : φ_A(x_1), φ_A(x_2) > α} is not a Cartesian product of two independent alpha-level sets unless the same membership condition is interpreted consistently. This is a minor presentational issue, as the intended reasoning is clear.
[Throughout] The paper would benefit from stating explicitly where the monotonicity of f is used and from identifying the exact condition under which the sum equality holds (e.g., f increasing). Currently the text gives the impression that the result holds for all continuous monotone injective f, which is misleading.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is an attempted derivation from stated lemmas; its failure is an invalid infimum/supremum decomposition, not a reduction to inputs.

full rationale

The paper is a mathematical derivation, not a fitting exercise. It defines f-correlated fuzzy numbers via a joint possibility distribution (Definition 1) and cites Lemmas 1–2 from previous work [6,7] as structural premises. Those lemmas are parameter-free and do not assert the theorem's conclusion; they only give the α-level formulas {x+f(x)} and {xf(x)} for the correlated operations. Theorem 1 then attempts to identify these sets with the standard interval sums and products. That identification is not circular in any of the listed senses: no parameter is fitted to data and then called a prediction, no known theorem is replaced by a self-citation, and the conclusion is not already contained in the definitions. The actual defect is the proof's step from Eq. (20) to Eq. (21), where inf/sup of x+f(x) is decomposed as inf x + inf f(x) and sup x + sup f(x); this holds only under special monotonicity and generally fails, e.g. f(x)=-x gives {0} but [L-U,U-L]. That is an invalid inference about interval endpoints, not a circular derivation. The paper's own Theorem 2 with f(x)=-x reveals the tension, but inconsistent theorems are a correctness problem, not evidence that a claim was assumed in its own proof. Self-citations [6,7] are load-bearing but normal mathematical citations to previously established lemmas, and nothing in the argument reduces to an unverified self-referential premise. Score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new entities or fitted parameters are introduced. The paper relies on standard fuzzy number theory and on the cited definition of f-correlated fuzzy numbers. The main unstated assumption is that the diagonal set {x + f(x)} can be treated as the Cartesian product of {x} and {f(x)}, which is false and causes the central error.

assumptions (4)

domain assumption Fuzzy numbers are identified with their alpha-level compact intervals.
Used implicitly throughout; standard in fuzzy arithmetic.
domain assumption The extension principle for interactive fuzzy numbers, Eq. (1), defines correlated operations.
Taken from the literature and used to derive Lemmas 1 and 2.
standard math Continuous monotone injective functions map intervals to intervals with endpoints f(a), f(b).
Used in Section 2 to compute alpha-levels of standard sums and products.
standard math The infimum and supremum of a sum of two sets equal the sum of the infima and suprema respectively.
Invoked in Theorem 1, but applied incorrectly to the diagonal set {x + f(x)}.

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Cite this review

Pith. "Pith review of A comparison of arithmetical operations with $f$ correlated fuzzy numbers." pith.science (2026). https://pith.science/paper/R7J3YDML

@misc{pith2026241200046,
  author       = {Pith},
  title        = {Pith review of: A comparison of arithmetical operations with $f$ correlated fuzzy numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7J3YDML}},
  note         = {Machine review of arXiv:2412.00046}
}

abstract

We present a brief introduction to a class of interactive fuzzy numbers, called $f$-correlated fuzzy numbers, which consist of pairs of fuzzy numbers where one is dependent on the other by a continuous monotone injective function. We have deduced some equations that can directly calculate the results of the sums and products of $f$-correlated fuzzy numbers, using only basic operations with real numbers, intervals on the real line and the function that relates the fuzzy numbers being considered. We proved that their correlated and standard sum coincide, and that in a certain sense, the correlated product is contained in the standard product.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

C., Bassanezi, R

Barros, L. C., Bassanezi, R. C., Lodwick, W . A. A ﬁrst course in fuzzy logic, fuzzy dynamical systems, and biomathematics: theory and applications. Springer, 2017

work page 2017

[2] [2]

C., Santo Pedro, F., Gomes, L

Barros, L. C., Santo Pedro, F., Gomes, L. T. ”Diferenc ¸as entre n´ umeros fuzzy.” Proceeding Series of the Brazilian Society of Computational and Applied Mathematic s, vol. 3, no. 1, 2015

work page 2015

[3] [3]

C., Santo Pedro, F

Barros, L. C., Santo Pedro, F. ”Fuzzy differential equat ions with interactive derivative.” Fuzzy Sets and Systems, vol. 309, pp. 64–80, 2017. doi:10.1016/j.fss.2016.04.00 2

work page doi:10.1016/j.fss.2016.04.00 2017

[4] [4]

”Generalized differentiabilit y of fuzzy-valued functions.” Fuzzy Sets and Systems , vol

Bede, B., Stefanini, L. ”Generalized differentiabilit y of fuzzy-valued functions.” Fuzzy Sets and Systems , vol. 230, pp. 119–141, 2013. doi:10.1016/j.fss.2012.10.0 03

work page doi:10.1016/j.fss.2012.10.0 2013

[5] [5]

Berberian, S. K. (1994). A ﬁrst course in real analysis . Springer. https://doi.org/10.1007/978-1-4419-8548-4

work page doi:10.1007/978-1-4419-8548-4 1994

[6] [6]

M., Prata, R

Cabral, V . M., Prata, R. A. C., Barros, L. C. ”Operac ¸ ˜ oes com n´ umeros fuzzy f-correlacionados.” Bio- matem´atica, vol. 23, pp. 15–32, 2013

work page 2013

[7] [7]

M., Prata, R

Cabral, V . M., Prata, R. A. C., & Barros, L. C. (2015). f-co rrelated fuzzy numbers applied to HIV model with protease inhibitor therapy. Mathware & soft computing: The Magazine of the European Soci ety for Fuzzy Logic and T echnology, 22(1), 46-51

work page 2015

[8] [8]

”Additions of c ompletely correlated fuzzy numbers.” 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat

Carlsson, C., Full´ er, R., Majlender, P . ”Additions of c ompletely correlated fuzzy numbers.” 2004 IEEE International Conference on Fuzzy Systems (IEEE Cat. No.04 CH37542), vol. 1, pp. 535–539, 2004. doi:10.1109/FUZZY .2004.1375791

arXiv 2004