Properties of f correlated fuzzy numbers

Diogo Sampaio da Silva; Roberto Antonio Cordeiro Prata

REVIEW 3 major objections 5 minor 14 references

Properties of f correlated fuzzy numbers

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read An f-correlated pair has every alpha-cut of one fuzzy number equal to the image under f of the other's alpha-cut, so complete correlation preserves triangular and trapezoidal shape.

desk verdict The main theorem isn't proven as written—it assumes the key equality it needs to derive, and the advertised general LR-shape preservation is only demonstrated in the linear case. read the letter →

arxiv 2412.00045 v1 pith:MN3N27O6 submitted 2024-11-23 math.GM

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

classification math.GM MSC 03E72

keywords fuzzynumbersf-correlatedinteractiveLR-typealpha-cutsshapepreservationcompletelycorrelatedextensionprinciple

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

The paper introduces f-correlated fuzzy numbers, a class of interactive fuzzy numbers whose joint possibility distribution is concentrated on the graph of a monotone injective function f. Its central claim is that f-correlation preserves LR-type shape: if A is an LR-type fuzzy number with injective shape functions and B is f-correlated to A, then every level set, or alpha-cut, of B is the image under f of the corresponding alpha-cut of A, so the endpoints of B's alpha-cuts are f applied to the endpoints of A's. In the linear special case of complete correlation, triangular and trapezoidal fuzzy numbers stay triangular and trapezoidal. The authors point toward applications in biomathematics, where interactive fuzzy numbers appear in models of dynamical systems.

What carries the argument

The argument rides on two pieces: the $\alpha$-cut representation of an LR-type fuzzy number, $[A]_{\alpha}=[q^{-}-aL^{-1}(\alpha), q^{+}+bR^{-1}(\alpha)]$ for injective shape functions L and R, and the interval-mapping property of continuous monotone injective functions, $f([a,b])=[f(a),f(b)]$ for increasing f with the order reversed for decreasing f. Together with the f-correlated joint possibility distribution, which charges only the graph $y=f(x)$, these turn membership-level information into endpoint-level equality: $[B]_\alpha=f([A]_\alpha)$.

What would settle it

Take A a triangular fuzzy number with support $[-1,1]$ and take $f:[0,1]\to[1,2]$, $f(x)=x+1$, which is monotone injective and continuous; since the paper does not require the support of A to lie in f's domain, for every $\alpha<1$ the interval $[A]_\alpha$ contains points outside $[0,1]$, so the expression $f([A]_\alpha)$ is undefined and the asserted equality $[B]_\alpha=f([A]_\alpha)$ cannot hold.

Watch

Extended reading notes

Core claim

The main result is Theorem 1: if A is an LR-type fuzzy number $A=(q^{-}, q^{+}, a, b)_{LR}$ with L and R injective, and B is f-correlated to A, then for every $\alpha\in[0,1]$ the $\alpha$-cut of B is $[B]_{\alpha}=f([A]_{\alpha})=f([q^{-}-aL^{-1}(\alpha), q^{+}+bR^{-1}(\alpha)])$. Because a continuous monotone injective function maps intervals to intervals, the endpoints of $[B]_\alpha$ are exactly the images of the endpoints of $[A]_\alpha$, with order preserved if f is increasing and reversed if f is decreasing. The paper reads this as a shape-preservation result: Corollary 1 states that if B is completely correlated to A, meaning f is affine, then B is also an LR-type fuzzy number whenever A is. Examples 4 and 5 verify that complete correlation sends triangular fuzzy numbers to triangular fuzzy numbers and trapezoidal fuzzy numbers to trapezoidal fuzzy numbers.

Load-bearing premise

The proof rests on the unstated premise that every level set, or alpha-cut, of B equals the image under f of the corresponding level set of A; that equality holds only when f is one-to-one, the support of A lies inside f's domain, and B's membership is exactly the membership of A carried along f.

Editorial extensions

If this is right

For any f-correlated pair satisfying Theorem 1, computing the alpha-cuts of B reduces to applying f to the endpoint formulas of A, making fuzzy interval arithmetic on such pairs a one-dimensional endpoint calculation.
In the completely correlated affine case $f(x)=qx+r$, alpha-cut endpoints transform affinely, so triangular fuzzy numbers remain triangular and trapezoidal fuzzy numbers remain trapezoidal under correlation.
Shape preservation means that a model starting with an LR-type fuzzy quantity and introducing an f-correlated partner does not need to switch to a different family of membership functions for the partner.
Because the result separates the effect of f from the shape functions L and R, checking whether an f-correlation is admissible is reduced to checking monotonicity and injectivity of f on the support of A.
For increasing f the ordering of lower and upper endpoints of $[B]_\alpha$ matches that of $[A]_\alpha$; for decreasing f the endpoints swap, which is all the information needed for interval arithmetic.

Reading between the lines

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

A natural extension beyond the paper's explicit statements is to construct the shape functions of B explicitly by composing f with the inverse shape functions $L^{-1}$ and $R^{-1}$; doing so would make the abstract's claim that B is also an LR-type fuzzy number fully explicit for general f.
The endpoint-map view suggests a testable consequence: in fuzzy differential equations with f-correlated pairs, replacing the joint distribution by the endpoint map f should reproduce the same alpha-cut evolution, which a simulation could check.
The affine case offers a template for other structured correlation families: for monotone injective parametric functions such as hyperbolic maps of the form $q/x+r$, analogous closed-form alpha-cut formulas should hold whenever the support of A avoids the singularity.
At the application level, f-correlation could propagate uncertainty through a nonlinear response f while preserving the qualitative triangular or trapezoidal shape of a prior, provided the support of A lies inside f's domain.

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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

Colleague, here's my take. The headline: the paper's main theorem, as stated, is not proved. The proof of Theorem 1 begins with [B]_alpha = f([A]_alpha) and never derives that equality from Definition 7. The abstract's broader claim—that f-correlation preserves LR-type shape for any monotone injective f—is never actually established. What is proven, correctly, is the linear/complete-correlation case.

Credit where it's due: the definitions are standard and clearly presented, and Examples 4 and 5 correctly work out the triangular and trapezoidal cases under complete correlation. The interval-mapping property for continuous monotone functions is used correctly. The citation of [4] for f-correlation is appropriate; the self-citation is background, not a circular move.

The soft spots are real. The missing equality is the load-bearing gap. It holds only if f is injective (assumed), the support of A lies in f's domain, and the marginal of the joint distribution for B is the pushforward of A along f. None of those conditions are stated. Even granting that equality, Theorem 1 only locates the endpoints of alpha-cuts; it doesn't show B itself is LR-type. That would require constructing new shape functions L' and R' from f, L, and R, and the paper supplies no such construction. Corollary 1 only handles linear f and has no separate proof. So the abstract overstates the result.

Minor issues: Example 1 contains typos in the trapezoidal membership function (the R branch is malformed), and the prose has a few slips. These don't change the mathematical idea but reduce precision.

Who is this for? Someone working on interactive fuzzy numbers might find the examples clarifying, but the novelty is low: Theorem 1 largely reduces to Definition 7 plus the cited alpha-cut property from [5]. I wouldn't cite it in my own work.

Recommendation: if this crossed my desk, I'd send it to review, but with a clear expectation of major revision. The underlying idea is plausible and the topic is legitimate, so it deserves referee time; but as written it is not acceptable. Ask the authors to prove the missing equality and either prove the general shape-preservation claim or explicitly restrict the theorem to the linear case.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces a class of interactive fuzzy numbers called f-correlated fuzzy numbers, where the joint possibility distribution is supported on the graph of a monotone injective function f. The main claimed result (Theorem 1) is that if A is an LR-type fuzzy number and B is f-correlated to A, then the alpha-cut endpoints of B are obtained by applying f to the alpha-cut endpoints of A. The paper also states a corollary for completely correlated (linear) fuzzy numbers and gives examples for triangular and trapezoidal fuzzy numbers. The abstract further claims that f-correlation preserves LR-type shape in general. The central proof is incomplete: the equality [B]_alpha = f([A]_alpha) is assumed rather than derived, and the abstract's general LR-type claim is not proven.

Significance. The notion of f-correlated fuzzy numbers is a natural generalization of completely correlated fuzzy numbers, and the endpoint-transformation formula, if properly proven, would be a useful tool for computations with interactive fuzzy numbers. The paper includes concrete examples for triangular and trapezoidal fuzzy numbers that are easy to follow. However, the advertised shape-preservation theorem for general nonlinear f is not established in the current version; the proof gap is repairable, but as written the main result is not fully supported. The contribution is modest and would require additional work on shape-function construction and domain conditions to be a complete paper.

major comments (3)

[Section 2.1, Theorem 1] The proof begins with the assertion that [B]_alpha = f([A]_alpha) for all alpha in [0,1], but this equality is neither derived from Definition 7 nor stated as a separate lemma. The equality requires the marginal compatibility condition phi_B(f(x)) = phi_A(x) for x in the support of A, which follows from Definition 7 only if the support of A is contained in the domain X of f and if one accounts for the fact that phi_B(y) = 0 for y not in f(X). These domain conditions are not stated in Theorem 1. Because the proof of the theorem relies entirely on this equality, the proof is incomplete as written.
[Abstract and Section 2.1] The abstract claims that if two fuzzy numbers are f-correlated and one is an LR-type fuzzy number, then the other is also an LR-type fuzzy number, for a general monotone injective f. Theorem 1, however, only locates the endpoints of the alpha-cuts of B as f applied to the endpoints of the alpha-cuts of A. It does not show that B admits an LR-type representation with some shape functions L' and R'. For an increasing f, one would need to construct L' and R' from f, L, and R (e.g., L'(t) = L(s) where t = [f(q-) - f(q- - as)]/[f(q-) - f(q- - a)]), and the paper supplies no such construction. The only cases where B is actually shown to be LR-type are the linear complete-correlation cases in Examples 4 and 5, and those examples cover only triangular and trapezoidal A. Thus the headline claim of the abstract is not established by the results in the paper.
[Corollary 1] Corollary 1 states that if A is an LR-type fuzzy number and B is completely correlated to A (linear f), then B is also an LR-type fuzzy number. No proof is given. The examples that follow cover only triangular and trapezoidal A, not general LR-type fuzzy numbers. A proof would need to handle both q > 0 and q < 0; in the decreasing case, the left branch of B uses the function R and the right branch uses L, so the conclusion that B is 'LR-type' requires either swapping the roles of L and R in the definition or redefining the shape functions. The corollary is therefore unsupported as stated.

minor comments (5)

[Example 1] The formula for R((u - q+)/b) is written as (q+ + b - u)/(q+ - u), which is negative on the interval [q+, q+ + b]; it should be (q+ + b - u)/b. Also, Definition 2 says that the closure of the support of A is [q- - a, q+ + beta], where beta should be b.
[Lemma 2] The proof uses the set comprehension with the strict inequality sup C(x) > alpha, whereas the alpha-level is defined by >= alpha. For fuzzy numbers with upper semi-continuous membership the two sets coincide, but the proof as written is not faithful to the definition.
[Section 2.1] There is a typo in 'remebember' (should be 'remember'). Also, the statement that continuous monotone injective functions map [a,b] onto [f(a), f(b)] requires that [a,b] be contained in the domain of f, which again connects to the missing domain conditions in Theorem 1.
[References] References [5] and [6] are duplicates of the same paper by Carlsson, Fuller, and Majlender; one should be removed.
[Example 3] In the hyperbolic case, the condition 0 notin [A]_0 is stated at the end, but it would be clearer to require that the support of A avoids 0 before defining the joint distribution, so that f is defined on the relevant set.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a direct but under-justified consequence of Definition 7, and the abstract's general LR-shape claim is a completeness gap, not a circular reduction.

full rationale

The paper does not fit parameters, predict from fitted data, or invoke a load-bearing self-citation chain. Definition 7 defines f-correlated fuzzy numbers through a graph-supported joint possibility distribution whose two expressions coincide on the graph, which entails the marginal compatibility phi_B(f(x)) = phi_A(x). Theorem 1's alpha-cut conclusion [B]_alpha = f([A]_alpha) is a one-line consequence of that definition together with continuity and injectivity of f; the proof, however, simply asserts this equality with 'Since for all alpha, [B]_alpha = f([A]_alpha)' instead of deriving it from Definition 7. That is an omitted justification, not circularity: the proof does not use the theorem as an independent input, and the equality is not a fitted or renamed quantity. The only self-citation is Definition 7 from [4], which includes a current co-author, but it supplies the background definition rather than an unverified uniqueness or existence result. Two real gaps are correctness/completeness issues rather than circularity: the abstract's claim that f-correlation preserves LR-type shape for general f is not established by Theorem 1, which only determines alpha-cut endpoints, and the general case would require constructing new shape functions that the paper does not provide; Corollary 1 covers only the linear completely-correlated case, and Examples 4 and 5 work the linear case explicitly. These omissions are real but do not make the derivation circular, because the central nontrivial claim is underproved rather than assumed or fitted. Under the stated criteria, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim uses no fitted numbers or new entities. It depends on standard alpha-level machinery and on the f-correlation definition from [4]. The key assumption that [B]_alpha = f([A]_alpha) is taken as given in the proof.

assumptions (5)

domain assumption The alpha-cut equality [B]_alpha = f([A]_alpha) holds for f-correlated fuzzy numbers with injective f.
Used as the first equality in the proof of Theorem 1 without derivation; it is a consequence of Definition 7 only under domain and injectivity conditions.
domain assumption The support of A is contained in the domain X of f.
Needed for f([A]_alpha) to be defined; not stated in Definition 7 or Theorem 1.
standard math LR-type fuzzy numbers are uniquely determined by their alpha-levels (Lemma 1).
Imported from [1] and used implicitly in Corollary 1 and the examples to move from alpha-levels back to membership functions.
standard math L and R are injective, hence invertible on their ranges.
Stated in Theorem 1; used to write the alpha-levels of A as [q- - a L^{-1}(alpha), q+ + b R^{-1}(alpha)].
standard math Continuous monotone injective functions map intervals to intervals.
Stated in Section 2.1 and used in Theorem 1.

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Cite this review

Pith. "Pith review of Properties of f correlated fuzzy numbers." pith.science (2026). https://pith.science/paper/MN3N27O6

@misc{pith2026241200045,
  author       = {Pith},
  title        = {Pith review of: Properties of f correlated fuzzy numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MN3N27O6}},
  note         = {Machine review of arXiv:2412.00045}
}

abstract

This paper presents some concepts of the theory of interactive fuzzy numbers, and mainly, a class of interactive fuzzy numbers, called $f$-correlated fuzzy numbers. We start from the foundations of general fuzzy mathematics and go through operations and the notion of interactivity for fuzzy numbers. The main result is that $f$-correlation preserve the shape of certains fuzzy numbers. More specificaly, if two fuzzy numbers are $f$ correlated, and one is a LR-type fuzzy number, the other is also a LR-type fuzzy number. This paper also presents some operations with the $f$-correlated fuzzy numbers wich are interesting to applications like biomathematics.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[4]

M., Prata, R

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

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Pith tools

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

[1] [4]

M., Prata, R

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

work page 2013

[2] [5]

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

Carlsson, C., Full´ er, R., Majlender, P . ”Additions of c ompletely correlated fuzzy numbers.” 2004 IEEE International Conference on Fuzzy Systems (IEEE C at. No.04CH37542), vol. 1, pp. 535–539, 2004. doi:10.1109/FUZZY .2004.13757 91

arXiv 2004

[3] [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 ap plications. Springer, 2017

work page 2017

[4] [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 Computationa l and Applied Mathe- matics, vol. 3, no. 1, 2015

work page 2015

[5] [3]

C., Santo Pedro, F

Barros, L. C., Santo Pedro, F. ”Fuzzy differential equat ions with inter- active derivative.” Fuzzy Sets and Systems , vol. 309, pp. 64–80, 2017. doi:10.1016/j.fss.2016.04.002

work page doi:10.1016/j.fss.2016.04.002 2017

[6] [6]

”Additions of comple tely correlated fuzzy num- bers.” 2004 IEEE International Conference on Fuzzy Systems (IEEE C at

Carlsson, C., Full´ er, R., others. ”Additions of comple tely correlated fuzzy num- bers.” 2004 IEEE International Conference on Fuzzy Systems (IEEE C at. No. 04CH37542), vol. 1, pp. 535–539, 2004

work page 2004

[7] [7]

”On additivity of the weighted possibilistic mean oper- ator.” 2013 IEEE 14th International Symposium on Computational In telligence and Informatics (CINTI) , pp

Coroianu, L., Full´ er, R. ”On additivity of the weighted possibilistic mean oper- ator.” 2013 IEEE 14th International Symposium on Computational In telligence and Informatics (CINTI) , pp. 303–308, 2013. 5

work page 2013

[8] [8]

”On multiplication of interac tive fuzzy numbers.” 2013 IEEE 11th International Symposium on Intelligent Systems a nd Informatics (SISY), pp

Coroianu, L., Full´ er, R. ”On multiplication of interac tive fuzzy numbers.” 2013 IEEE 11th International Symposium on Intelligent Systems a nd Informatics (SISY), pp. 181–185, 2013

work page 2013