Relationships among three types of covering rough sets

Rough sets, a technique of granular computing, deal with the vagueness and granularity in information systems. They are based on equivalence relations on a set, or equivalently, on a partition on the set. Covering is an extension of a partition and a more feasible concept for coping with incompleteness in information, thus the classical rough sets based on partition are extended to covering based rough sets. When a covering is introduced, there are more than one possibility to define the upper approximation. It is necessary to study the properties of these different types of upper approximations and the relationships among them. This paper presents three kinds of covering generalized rough sets and explores the relationships among them. The main results are conditions under which two different types of upper approximation operations are identical.