On rough approximations via ideal

In this paper, we generalize the rough set model by defining new approximation operators in more general setting of a complete atomic Boolean lattice by using an ideal. An ideal on a set X is a non empty collection of subsets of X with heredity property which is also closed under finite unions. We introduce the concept of lower and upper approximations via ideal in a lattice theoretical setting. These decrease the upper approximation and increase the lower approximation and hence increase the accuracy. Properties of these approximations are studied. Also properties of the ordered set of the lower and upper of an element of a complete atomic Boolean lattice via ideal are investigated. We also study the connections between the rough approximations defined by Järvinen and our new approximations. Various examples are given. Finally we give a new approach for defining the rough approximations w.r.t the induced map by using an ideal. We study the connections between the rough approximations defined with respect to the induced map by using an ideal and the rough approximations defined with respect to the considered map under certain conditions of the map.