Partial Approximative Set Theory: A View from Galois Connections

The rough set theory was introduced by the Polish mathematician, Z. Pawlak in the early 1980s. It was a new mathematical approach to vagueness. His idea can be formulated in two different ways, namely, in a point-wise and in a point-free manner. Its points-wise generalization can be described, e.g., on the language of relation algebra, and it is a well-known th eory. In this paper, our starting point will be an arbitrary family of subsets of the universe. Neither that it covers the universe nor that the universe is finite will be assumed. Moreover, within this framework, our concepts of lower and upper approximations are straightforward point-free generalizations of Pawlak’s ones. We shall investigate by what conditions such so-called weak lower and upper approximations form Galois connections. We will compare our result to the similar result given by the point-wise gene ralization.