  Fuzzy rough set (FRS) has a great effect on data mining processes and the
fuzzy logical operators play a key role in the development of FRS theory. In
order to further generalize the FRS theory to more complicated data
environments, we firstly propose four types of fuzzy neighborhood operators
based on fuzzy covering by overlap functions and their implicators in this
paper. Meanwhile, the derived fuzzy coverings from an original fuzzy covering
are defined and the equalities among overlap function-based fuzzy neighborhood
operators based on a finite fuzzy covering are also investigated. Secondly, we
prove that new operators can be divided into seventeen groups according to
equivalence relations, and the partial order relations among these seventeen
classes of operators are discussed, as well. Go further, the comparisons with $
t$-norm-based fuzzy neighborhood operators given by D'eer et al. are also made
and two types of neighborhood-related fuzzy covering-based rough set models,
which are defined via different fuzzy neighborhood operators that are on the
basis of diverse kinds of fuzzy logical operators proposed. Furthermore, the
groupings and partially order relations are also discussed. Finally, a novel
fuzzy TOPSIS methodology is put forward to solve a biosynthetic nanomaterials
select issue, and the rationality and enforceability of our new approach is
verified by comparing its results with nine different methods.
