Bi-Fuzzy S-Approximation Spaces

The S-approximation spaces are significant extension of the rough set model and have been widely applied in intelligent decision-making. However, traditional S-approximation spaces are limited to two crisp universes, whereas bi-fuzzy universes (i.e., two distinct fuzzy domains) are more prevalent in practical applications. To bridge this gap, this study introduces the bi-fuzzy S-approximation spaces (BFS approximation spaces) as an advancement of knowledge space theory’s fuzzy extension. Upper and lower approximation operators are formally defined, and the properties of BFS approximation spaces under various operations, such as complement, intersection and union are systematically explored. Special attention is given to a significant form of these operators, under which the monotonicity and complementary compatibility of BFS approximation spaces are rigorously analyzed. These results not only extend the theoretical framework of S-approximation spaces but also pave the way for further exploration of fuzzy extensions within knowledge space theory.