Some Algebraic Properties of Multigranulations and an Analysis of Multigranular Approximations of Classifications

Ever since the introduction of rough sets by Pawlak as a model to capture uncertainty, it has drawn much attention from both theoretical and application point of view. Classifications of universes play very important roles in several fields of study. The study of rough definability of classifications was initiated by Busse. The properties of approximations of classifications were established in the form of four theorems and were used to define the types of classifications. These results were generalised to develop two theorems of necessary and sufficient type were established by Tripathy et al , from which several results including the four theorems of Busse could be derived as corollaries. Recently, rough sets based on Multigranulation were introduced and studied by Qian et al. Also, it has been extended to include incomplete information systems. Many of these results are extended to the multigranular cases. In this paper, we extend the properties of types of classifications to the multigranular context. Also, we introduce some parameters like the accuracy of approximation and the quality of approximation of classifications with respect to Multigranulations. We have obtained interesting criteria under which both types of Multigranulations reduce to single granulation. Also, some algebraic properties of Multigranulations are derived.