Approximation of Sets Based on Partial Covering

In classic Pawlakian rough set theory the sets used to approximations are equivalence classes which are pairwise disjoint and cover the universe. In this article we give up not only the pairwise disjoint property but also the covering of the universe.

After a historical and philosophical background, we define a general set theoretic approximation framework. First, we reconstruct the rough set theory and partly restate its some well–known facts in the language of this framework.

Next, we present a special approximation scheme. It is based on the partial covering of the universe which is called the base system and denoted by

$\mathfrak{B}$

.

$\mathfrak{B}$

-definable sets and lower and upper

$\mathfrak{B}$

-approximations are straightforward point–free generalizations of Pawlakian ones. We study such notions as single–layered base systems,

$\mathfrak{B}$

-representations of

$\mathfrak{B}$

-definable sets, and the exactness of sets. It is a well–known fact that the Pawlakian upper and lower approximations form a Galois connection. We clarify which conditions have to be satisfied by the upper and lower

$\mathfrak{B}$

-approximations so that they form a (regular) Galois connection. Excluding the cases when the empty set is the upper

$\mathfrak{B}$

-approximation of certain nonempty sets gives rise to a partial upper

$\mathfrak{B}$

-approximation map. We also clear up that a partial upper

$\mathfrak{B}$

-approximation map and a total lower

$\mathfrak{B}$

-approximation map form a partial Galois connection.

In order to demonstrate the effectiveness of our approach we present three real–life examples in the last section.