Partial order relation for approximation operators in covering based rough sets

Covering based rough sets are a generalization of classical rough sets, in which the traditional partition of the universe induced by an equivalence relation is replaced by a covering. Many definitions have been proposed for the lower and upper approximations within this setting. In this paper, we recall the most important ones and organize them into sixteen dual pairs. Then, to provide more insight into their structure, we investigate order relationships that hold among the approximation operators. In particular, we study a point-wise partial order for lower (resp., upper) approximation operators, that can be used to compare their respective approximation fineness. We establish the Hasse diagram for the partial order, showing the relationship between any pair of lower (resp., upper) operators, and identifying its minimal and maximal elements.