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Granular computing is a widely used computational paradigm nowadays. Particularly, within the rough set theory, granular computing plays a key role. In this paper, we propose a generic approach of rough sets, the granular extended multigranular sets (GEMS) for dealing with both mixed and incomplete information systems. Not only our proposal does use the traditional optimistic and pessimistic granulations with respect to single attributes, but also we introduce granulations with respect to attribute sets, as well as two new ways of granulating: the optimistic + pessimistic granulation and the pessimistic + optimistic granulation. In addition, we have developed a particular case of the proposed GEMS: the multigranular maximum similarity rough sets (MMSRS). We have proved some of the properties of the MMSRS, and we tested its effectiveness with respect to other existing granular rough sets models. The experimental results show the flexibility and the capabilities of the proposed model, while handling mixed and incomplete information systems. [ABSTRACT FROM AUTHOR]