A novel notion in rough set theory: Invariant subspace

All current work on calculating approximate sets of a rough set inevitably requires the participation of all elements in the universe. However, it is cumbersome and causes a huge waste of resources when the universe is quite big and the computed set is small enough. By introducing the notion of invariant subspace in rough set theory, we skillfully reduce the range of elements involved in the computations of approximations of a rough set from the whole universe U to a suitable invariant subspace V of U, and then give the modular method within the reduced range. Moreover, we present the modular Boolean matrix method, such that the calculation of upper and lower approximations of rough sets can be converted into operations on modular matrices. Finally, the results in covering approximation space are generalized to fuzzy β-covering approximation space to calculate the upper and lower approximations of the crisp sets. In particular, an algorithm for calculating the value of β is proposed to make the involved information has the highest distinction degree. This β is more reliable and meaningful than the empirical one, and an example about COVID-19 is put forward to simply illustrate its application.