Decision reducts and bireducts in a covering approximation space and their relationship to set definability

In this paper, we discuss the relationship between different types of reduction and set definability. We recall the definition of a decision reduct, a γ-decision reduct, a decision bireduct and a γ-decision bireduct in a Pawlak approximation space and the notion of set definability both in a Pawlak and a covering approximation space. We extend the notion of discernibility between objects in a Pawlak approximation space to a covering approximation space. Moreover, we introduce the definition of a decision reduct, a γ-decision reduct, a decision bireduct and a γ-decision bireduct in a covering approximation space. In addition, we study interrelationships between the four types of reduction, the correspondence with positive regions and the relationship to set definability in Pawlak and covering approximation spaces.