In this paper we address the problem of
α
-migrativity (for a fixed
α
) for semicopulas, copulas and quasi-copulas. We introduce the concept of an
α
-sum of semicopulas. This new concept allows us to ...completely characterize
α
-migrative semicopulas and copulas. Moreover,
α
-sums also provide a means to obtain a partial characterization of
α
-migrative quasi-copulas.
In this paper, we study convergence in measure theorems without the continuity of the measure with respect to seminormed and semiconormed fuzzy integrals. By using the obtained results, we achieve ...results about pointwise convergence and almost uniform convergence for seminormed fuzzy integrals.
In this paper, we introduce a new property of a semicopula, called the uniform left (or right)-continuity in the first (or second) variable. Based on this new concept of continuity, a uniform ...convergence theorem for the smallest semicopula-based universal integral is given. In particular, a counter-example is presented to show that Theorem 2.9 in Borzová-Molnárová et al. (2015) 4 is not true. Finally, some modified versions of Theorems 2.7, 2.8 and 2.9 in Borzová-Molnárová et al. (2015) 4 are studied.
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.
In this paper, we study two properties of the seminormed fuzzy integral. By applying these results, we propose alternative proof of the monotone convergence theorems for smallest semicopula-based ...universal integrals, which are proposed by J. Borzová-Molnárová et al. in 2015.
Semicopula based integrals Jin, LeSheng; Kolesárová, Anna; Mesiar, Radko
Fuzzy sets and systems,
06/2021, Letnik:
412
Journal Article
Recenzirano
We define the notion of weak universal integral based on a semicopula, and introduce and discuss two particular classes of weak universal integrals based on semicopulas, which generalize the ...well-known Sugeno and Shilkret integrals. In special cases, when the considered semicopulas are bounded from above by the Łukasiewicz t-norm, all introduced integrals reduce to the corresponding smallest semicopula based universal integrals. Remarkably, when the product semicopula is considered, the proposed integrals generalizing the Shilkret integral belong to the class of aggregation functions, which is not the case of the minimum semicopula when the introduced integrals generalize the Sugeno integral.
The aim of this paper is mainly to solve the functional equations given by the modularity condition. Several years ago, the modularity equations for t-norms, t-conorms, uninorms and t-operators, ...which are commutative and associative, have been studied. Our investigations are motivated by modularity condition for generalizations of these operators by removing associativity or commutativity. In this work, the following main conclusions are proved: (1) a continuous t-norm with respect to a continuous semicopula is modular if and only if they are equal. The case for a semicopula with respect to a strict t-norm is also the same. A semicopula with respect to a co-semicopula is modular if and only if the semicopula is min and the co-semicopula is max. The modularity condition does not hold for a co-semicopula with respect to a semicopula. (2) Necessary and sufficient conditions are given for a semi-t-operator with respect to a semi-uninorm, a pseudo-uninorm with respect to a semi-t-operator to satisfy the modularity condition equation. New solutions to the modularity condition equations of the Case (1) are characterized.
Semicopulas are the operators chosen to model conjunction in the fuzzy/many-valued logics. In fact, a special kind of semicopula, called t-norm, is widely used in many applications of logic to ...engineering, computer science and fuzzy systems. The main result of this paper is the computation of the exact number of semicopulas that can be defined on a finite chain in terms of its length. The final formula is achieved via relating semicopulas with finite plane partitions.
In this paper, we deal with the transformation theorem for the generalized upper Sugeno integral I∘. We formulate the sufficient and necessary conditions under which the transformation holds, thus ...providing a complete solution to this problem. We also provide some representations of the I∘ integral via left-continuous (right-continuous) modifications of level measures. We characterize the class of binary operations for which the I∘ integral can be represented via the quantile function. As a consequence, we get complete solutions to these problems for the smallest semicopula-based universal integral. We thus solve several open problems related to this class of integrals in the literature.
