On characterization of fuzzy soft rough sets based on a pair of border implicators.

*(English)*Zbl 1357.68240Summary: Fuzzy set theory, soft set theory and rough set theory are powerful mathematical tools for dealing with various types of uncertainty. This paper is devoted to define a broad family of soft fuzzy roughsets, each one of which, called an \((I, J)\)-soft fuzzy rough set, is determined by a pair of border implicators \((I, J)\). Alternatively, it shows that a fuzzy soft set can induce a \(T\)-equivalence fuzzy relation which is used to granulate the universe. In particular, we prove that \((I, J)\)-fuzzy soft rough sets in our work are equivalent to \((I, J)\)-fuzzy rough sets of Y. Ouyang et al. [Inf. Sci. 180, No. 4, 532–542 (2010; Zbl 1189.68131)] by using a \(T\)-equivalence fuzzy relation determined by a fuzzy soft set. Furthermore, basic properties of \((I, J)\)-fuzzy soft rough sets are investigated. Meanwhile, an operator-oriented characterization of \((I, J)\)-fuzzy soft rough sets is proposed. Finally, an example is given to illustrate the approach of present paper.

##### MSC:

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

03E72 | Theory of fuzzy sets, etc. |