Abstract
Rough set theory was introduced by Pawlak, in 1982, as a methodology to discover structural relationships within imprecise and uncertain data. This theory has been generalized using the idea of neighborhood systems to be more efficient to get rid of uncertainty and deal with a wide scope of practical applications. Motivated by this idea, in this work, we initiate novel generalized rough set models using the concepts of "maximal left neighborhoods and ideals." Their basic features are studied and the relationships between them are revealed. The main merits of these models, as we prove, are first to preserve almost all major properties of approximation operators with respect to the Pawlak model. Second, they keep the monotonic property, which leads to an efficient evaluation of the uncertainty in the data, and third, these models enlarge the knowledge gotten from the information systems because they minimize the vagueness regions more than some previous models. We complete this manuscript by applying the proposed approach to analyze educational data and illustrate its role to improve the obtained classifications of objects and show the great performance of the present approach against other ones. Elucidative examples that support the obtained results are provided.