Relationship among basic concepts in covering-based rough sets

Uncertainty and incompleteness of knowledge are widespread phenomena in information systems. Rough set theory is a tool for dealing with granularity and vagueness in data analysis. Rough set method has already been applied to various fields such as process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, and conflict analysis. Covering-based rough set theory is an extension to classical rough sets. In covering-based rough sets, there exist several basic concepts such as reducible elements of a covering, minimal descriptions, unary coverings, and the property that the intersection of any two elements is the union of finite elements in this covering. These concepts appeared in the literature of covering-based rough sets separately. In this paper we study the relationships between them. In particular, we establish the equivalence of the unary covering and the covering with the property that the intersection of any two elements is the union of finite elements in this covering. We also investigate the relationship between the covering lower approximation operation and the interior operator. A characterization of the interior operator by the covering lower approximation operation is presented in this paper. Correspondingly, we study the relationship between the covering upper approximation operation and the closure operator. In addition, we explore the conditions under which the covering upper approximation operation is monotone. The study of the relationships between these concepts will help us have a better understanding of covering-based rough sets.

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