From comonotone commuting properties of seminormed fuzzy integrals to solving two open problems
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 equalityholds for any two comonotone functions , any fuzzy measure space and any . They proved that if and for all , then Sugeno integral possesses the comonotone -commuting property if and only if is one of four operators: and . This result is also true for the integral provided that and for all [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 be fixed. Find all operators such that for any measurable space , any nondecreasing measure and any for all comonotone functions , where and stands for the indicator of A.
In case , 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 of all increasing in the both coordinates, associative and continuous semicopulas, and some class of binary operators .
Motivated by these issues, we extensively examined both Problem 1 and 2. The outstanding results in our paper are the following points.
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If is a solution of the open problems then is nondecreasing (see Theorem 3).
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If a semicopula then it is impossible a semicopula solution of the open problems. In particular, if then Problem 2 only possess a semicopula solution that is itself (see Proposition 7, Proposition 8).
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From a simple condition for each semicopula it follows that if is a solution of the open problems then is continuous on (see Proposition 20)
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Our work gives the solution of Problem 1 and 2 in 10 classes of binary operators and the class of semicopulas (see Theorem 7, Theorem 12). Further, it also gives some cases of semicopulas and binary operators such that is not the solution of the open problems (see Theorem 8).
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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 be a measurable space, where is a -algebra of subsets of a nonempty set X. Definition 1 [[13]] Let be a non-negative, extended real-valued set function. is called a monotone measure if it satisfies . and imply .
Then the triplet is called a monotone measure space.
Let , we denote by the -level set of f for . Definition 2 Let and
Main result
This section is devoted to characterize the solution of Problem 2. Studying of this problem is divided into two main ideas.
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Some necessary and sufficient conditions for a binary operator belonging to is stated.
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The characterization for the solution of Problem 2 are specified.
First, we give an important necessary condition leads to an operator belongs to . Theorem 3 For each . If then is nondecreasing. Proof It is sufficient to prove that and for all with . Now,
A relationship between Problem 1 and 2
First, we have the following a special property of all solutions of Problem 1. Proposition 22 i.e., and . Proof It is clear that . Now, let for all . By taking any comonotone functions and . Then we have This shows that . On the other hand, by choosing with one hasThis implies that . So, . The proof is finished.
Next, another special property of is given. Proposition 23 For each .
Conclusion
Our results are remarkable improvements to ones that had been previously studied.
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Theorem 2 is a perfect supplement for solving the open problem 9.3 in [2], [5], [8].
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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).
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In Theorem 4, we shown that the form of binary operators being solutions of Problem 1 contains an additional constant which need not vanish.
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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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