Duality, conjugacy and adjointness of approximation operators in covering-based rough sets

• We establish correspondences between covering-based approximation operators.
• We highlight adjointness as a complementary property to duality.
• We evaluate existing operators with respect to adjointness.
• We present a characterization of dual and adjoint pairs in terms of conjugacy.
• We relate our results to generalized rough sets based on a binary relation.

Many different proposals exist for the definition of lower and upper approximation operators in covering-based rough sets. In this paper, we establish relationships between the most commonly used operators, using especially concepts of duality, conjugacy and adjointness (also referred to as Galois connection). We highlight the importance of the adjointness condition as a way to provide a meaningful link, aside from duality, between a pair of approximation operators. Moreover, we show that a pair of a lower and an upper approximation operator can be dual and adjoint at the same time if and only if the upper approximation is self-conjugate, and we relate this result to a similar characterization obtained for the generalized rough set model based on a binary relation.