Double-quantitative $\gamma^{\ast}-$fuzzy coverings approximation
  operators
  In digital-based information boom, the fuzzy covering rough set model is an
important mathematical tool for artificial intelligence, and how to build the
bridge between the fuzzy covering rough set theory and Pawlak's model is
becoming a hot research topic. In this paper, we first present the
$\gamma-$fuzzy covering based probabilistic and grade approximation operators
and double-quantitative approximation operators. We also study the
relationships among the three types of $\gamma-$fuzzy covering based
approximation operators. Second, we propose the $\gamma^{\ast}-$fuzzy coverings
based multi-granulation probabilistic and grade lower and upper approximation
operators and multi-granulation double-quantitative lower and upper
approximation operators. We also investigate the relationships among these
types of $\gamma-$fuzzy coverings based approximation operators. Finally, we
employ several examples to illustrate how to construct the lower and upper
approximations of fuzzy sets with the absolute and relative quantitative
information.
