Elsevier

Information Sciences

Volume 574, October 2021, Pages 33-50
Information Sciences

From comonotone commuting properties of seminormed fuzzy integrals to solving two open problems

https://doi.org/10.1016/j.ins.2021.05.082Get rights and content

Abstract

In this paper, we characterize the class of solutions for two open problems: one is Problem 1 which is posed by Ouyang et al. (2009), other is Problem 2 which is proposed by Borzová-Molnárová et al. (2015). Many results being wider than the previous ones have been stated. All are summarized in Theorems: 2,3,4,5,7,9,11 and 12.

Introduction

Sugeno integral is one of the important tools for modeling non-deterministic problems [6]. One of the prominent issues related to Sugeno integral is comonotone commuting property [10], [11], [12]. After works on the Chebyshev inequality for Sugeno integral and two comonotone functions with respect to a binary operator [10], Ouyang et al. was particularly interested in the following problem.

Determine when the equality(S)Afgdμ=(S)Afdμ(S)Agdμholds for any two comonotone functions f,g, any fuzzy measure space X,A,μ and any AA. They proved that if limbaba, and limaabb, for all a,b, then Sugeno integral possesses the comonotone -commuting property if and only if is one of four operators: ,,PF and PL. This result is also true for the integral IMμ,f=sup0<α1αμfα provided that limb1aba,1 and lima1abb,1 for all a,b0,1 [11]. Moreover, they also proved that there are only 18 operators such that the equality (1) holds for all measurable and comonotone functions. Inspired by the obtained result, they posed the following open problem.

Problem 1

Let SS be fixed. Find all operators :0,120,1 such that for any measurable space X,A, any nondecreasing measure μMX,A1 and any AAISμ,fgA=ISμ,fAISμ,gAfor all comonotone functions f,g:X0,1, where hA=hχA and χA stands for the indicator of A.

In case A=X, we obtain Problem 2 which was formulated by Borzová-Molnárová et al. [2], when they had studied linearity, homogeneity and maxity of the smallest semicopula-based universal integral.

Afterward, Boczek et al. [4] studied Problem 1. In Corollary 1, they pointed out all binary operators that are comonotone commuting for Sugeno integral. Further, in Corollary 2 they also give the solution of Problem 1 in the class S of all increasing in the both coordinates, associative and continuous semicopulas, and some class of binary operators :0,120,1.

Motivated by these issues, we extensively examined both Problem 1 and 2. The outstanding results in our paper are the following points.

  • If is a solution of the open problems then is nondecreasing (see Theorem 3).

  • If a semicopula SMthen it is impossible a semicopula solution of the open problems. In particular, if S=M then Problem 2 only possess a semicopula solution that is M itself (see Proposition 7, Proposition 8).

  • From a simple condition for each semicopula it follows that if is a solution of the open problems then is continuous on 0,12 (see Proposition 20)

  • Our work gives the solution of Problem 1 and 2 in 10 classes of binary operators and the class S of semicopulas S (see Theorem 7, Theorem 12). Further, it also gives some cases of semicopulas S and binary operators such that is not the solution of the open problems (see Theorem 8).

  • Our paper clearly points out the relationship between Problem 1 and 2. From that, Problem 1 is also dealt with through studying Problem 2 (see Theorem 9).

The organization of the paper: in Section 2, we recall the background of the smallest semicopula-based universal integral and related results. In Section 3, Problem 2 is solved. In Section 4, the relationship of Problem 1 and 2 is presented. Next, a conclusion is given in Section 5. Finally, Appendix is given in Section A.

Section snippets

Preliminaries

In this section, we recall and introduce some necessary concepts. Further, we also show some results that support Section 3.

Let X,A be a measurable space, where A is a σ-algebra of subsets of a nonempty set X.

Definition 1

[[13]] Let μ:A0, be a non-negative, extended real-valued set function. μ is called a monotone measure if it satisfies

  • 1.

    μ=0.

  • 2.

    A,BAand AB imply μAμB.

Then the triplet X,A,μ is called a monotone measure space.

Let f:X0,, we denote by fα=xXfxα the α-level set of f for α>0.

Definition 2

Let f,g:X0,1 and

Main result

This section is devoted to characterize the solution of Problem 2. Studying of this problem is divided into two main ideas.

  • Some necessary and sufficient conditions for a binary operator belonging to PS is stated.

  • The characterization for the solution of Problem 2 are specified.

First, we give an important necessary condition leads to an operator belongs to PS.

Theorem 3

For each SS. If PS then is nondecreasing.

Proof

It is sufficient to prove that ab1ab2 and b1ab2a for all a,b1,b20,1 with b1b2. Now,

A relationship between Problem 1 and 2

First, we have the following a special property of all solutions of Problem 1.

Proposition 22

PSPS i.e., PSPS and PSPS.

Proof

It is clear that PSPS. Now, let ab=1 for all a,bX. By taking any comonotone functions f,gFX,A0,1 and μMX,A1. Then we haveS,Xfgdμ=S1,μX=1=S,XfdμS,Xgdμ.

This shows that PS. On the other hand, by choosing AA with μA=12 one hasS,Afgdμ=S1,μA=12<S,AfdμS,Agdμ.This implies that PS. So, PSPS. The proof is finished.

Next, another special property of PSis given.

Proposition 23

For each SS.

Conclusion

Our results are remarkable improvements to ones that had been previously studied.

  • Theorem 2 is a perfect supplement for solving the open problem 9.3 in [2], [5], [8].

  • Our work does not only characterize the solutions of Problem 1 but also Problem 2. Further, the relationship between Problem 1 and 2 is clearly pointed out (see Theorem 9).

  • In Theorem 4, we shown that the form of binary operators being solutions of Problem 1 contains an additional constant e0=00 which need not vanish.

  • Lemma 1 is

CRediT authorship contribution statement

Tran Nhat Luan: Writing - original draft, Validation, Writing - review & editing, Methodology. Do Huy Hoang: Writing - review & editing. Tran Minh Thuyet: Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors would like to express deeply gratitude to the Editor-in-Chief: Professor Witold Pedrycz and the anonymous referees for their valuable comments and suggestions which have greatly improved this paper.

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