On two new types of fuzzy rough sets via overlap functions and corresponding applications to three-way approximations

Fuzzy rough sets (FRSs) can be served as a useful technique to cope with inconsistent data by providing a pair of lower and upper fuzzy rough approximations of a fuzzy concept. Diversified FRS models have been proposed over the years. Meanwhile, t-norms and t-conorms, as two binary aggregation operators, play a critical part in the construction of fuzzy rough approximation operators. Nevertheless, one of common properties of t-norms and t-conorms is that they all fulfill associativity. Additionally, the associativity plays an important role in discussing properties of FRSs. Therefore, if the binary aggregation function does not satisfy associativity, how to define FRS models and discuss the relevant properties is a topic worthy of further investigation. Overlap functions, as a new kind of not necessary associative binary aggregation operator, have been continuously investigated by many scholars in theory and practical applications. Therefore, based on the advantages of overlap functions, and as the extension models for FRSs, we propose two new types of FRSs via overlap functions and their residual implications (or called I O -implications) in this paper. More specifically, we first give the definition of the 1 st type of FRSs and study some essential properties. Subsequently, the definition of the 2 nd type of FRSs is presented and the relevant properties are also investigated. Furthermore, the relationship between the 1 st type of FRSs and the 2 nd type of FRSs is described. That is, in a fuzzy approximation space, the 2 nd type of lower approximation is contained in the 1 st type of lower approximation, while the 1 st type of upper approximation is included in the 2 nd type of upper approximation when the overlap function has 1 as the identity element. At the end, a novel three-way decision (3WD) method is presented on an information table by using the new type of FRSs, and its practicality and superiority are verified via a numerical example and a series of experimental analyses.