Generalized Interval-Valued Fuzzy Rough Sets Based on Interval-Valued Fuzzy Logical Operators

The fuzzy rough sets and generalized fuzzy rough sets have been extended by three pairs of fuzzy logical operators to deal with real-valued data for a variety of models. Three pairs of fuzzy logical operators are triangular norm or t-norm and its dual (t-conorm), residual implicator or R-implicator and its dual, fuzzy implicator and t-norm, which are frequently discussed in generalization models of rough sets. There are recent researches into generalized interval-valued fuzzy rough sets for interval-valued fuzzy datasets. However, only fuzzy implicator and t-norm are used in generalized interval-valued fuzzy rough sets and another two pairs of fuzzy logical operators, t-norm and t-conorm, R-implicator and its dual, have not been considered in generalized interval-valued fuzzy rough sets. In this paper, these models are generalized to a new approach that not only considers interval-valued fuzzy sets but also two pairs of fuzzy logical operators. First, we study the interval representations of interval-valued fuzzy t-norm, interval-valued fuzzy negator, interval-valued fuzzy R-implicator and their duals, which provide a theoretical basis for interval-valued fuzzy rough sets based on interval-valued fuzzy logical operators. Second, we propose generalized interval-valued fuzzy rough sets based on two pairs of fuzzy logical operators: interval-valued fuzzy t-norms and t-conorm, and interval-valued fuzzy R-implicators and its dual. Finally, we confirm that some existing models, including rough sets, interval-valued fuzzy rough sets and generalized fuzzy rough sets based on fuzzy logical operators are special cases of the proposed models.