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Aggregation and related functions in fuzzy set theory: dualities and monotonicity-based generalizations

Mesiar R., Kolesarova A.

Journal of Automation Mobile Robotics and Intelligent Systems

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2017

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Vol. 11, No. 3

58--61

EN

We discuss several extensions of binary Boolean functions acting on the domain [0, 1]. Formally, there are 16 disjoint classes of such functions, covering a majority of binary functions considered in fuzzy set theory. We introduce and discuss dualities in this framework, stressing the links between different subclasses of considered functions, e.g., the link between conjunctive and implication functions. Special classes of considered functions are characterized, among others, by particular kinds of monotonicity. Relaxing these constraints by considering monotonicity in one direction only, we generalize standard classes of aggregation functions, implications, semicopulas, etc., into larger classes called pre-aggregations, pre-implications, pre-semicopulas, etc. Note that the dualities discussed for the standard classes also relate the new extended classes of pre-functions.

2

Approximation-oriented Fuzzy Rough Set Approaches

Inuiguchi M.

Fundamenta Informaticae

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2015

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Vol. 142, nr 1-4

21--51

EN

In this paper we focus on generalizations of the classical rough set approach to fuzzy environments. There are two aspects of rough set approaches: classification and approximation. In the classification aspect, by rough set approaches we can classify objects into positive and negative examples of a class. On the other hand, in the approximation aspect, by rough set approaches we obtain the lower and upper approximations of a class. The former model works better in the attribute reduction while the latter model works better in the rule induction. In the setting of the classical rough set approach, the lower approximation is nothing but the set of positive examples and the upper approximation is the complementary set of negative examples. However, these equalities do not always hold in the generalized settings. Most of fuzzy rough set models proposed earlier are defined in the classification aspect. The approaches based on those models do not always work well in approximating fuzzy subsets. In this paper we define the fuzzy rough set models in the approximation aspect. We investigate their fundamental properties and demonstrate the advantages of fuzzy set approximation. Finally we consider attribute reduction based on the proposed fuzzy rough set models.

3

Necessity measures and parametric inclusion relations of fuzzy sets

Inuiguchi M., Tanino T.

Fundamenta Informaticae

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2000

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Vol. 42, nr 3,4

279-302

EN

A necessity measure N is defined by an implication function. However, specification of an implication function is difficult. Necessity measures are closely related to inclusion relations. In this paper, we propose an approach to necessity measure specification by giving a parametric inclusion relation between fuzzy sets A and B which is equivalent to NA(B) > h. It is shown that, in such a way, we can specify a necessity measure, i.e., an implication function. Moreover, when a necessity measure or equivalently, an implication function is given, then the derivation of an associated parametric inclusion relation is discussed. The associated parametric inclusion relation cannot be obtained for any implication function but only for implication functions which satisfy certain conditions. Applying our results to necessity measures defined by S-, R- and reciprocal R-implications with continuous Archimedean t-norms, associated parametric inclusion relations are shown.