The notions of basic rough sets introduced by Pawlak as a model of uncertainty, which depends upon a single equivalence relation has been extended in many directions. Over the years, several extensions to this rough set model have been proposed to improve its modeling capabilities. From the granular computing point of view these models are single granulations only. This single granulation model has been extended to multi-granulation set up by taking more than one equivalence relations simultaneously. This led to the notions of optimistic and pessimistic multi-granulation. One direction of extension of the basic rough set model is dependent upon covers of universes instead of partitions and has better modeling power as in many real life scenario objects cannot be grouped into partitions but into covers, which are general notions of partitions. So, multigranulations basing on covers called covering based multi-granulation rough sets (CBMGRS) were introduced. In the literature four types of CBMGRSs have been introduced. The first two types of CBMGRS are based on minimal descriptor and the other two are based on maximal descriptor. In this paper all these four types of CBMGRS are studied from their topological characterizations point of view. It is well known that there are four kinds of basic rough sets from the topological characterisation point of view. We introduce similar characterisation for CBMGRSs and obtained the kinds of the complement, union, and intersection of such sets. These results along with the accuracy measures of CBMGRSs are supposed to be applicable in real life situations. We provide proofs and counter examples as per the necessity of the situations to establish our claims.

The notion of rough set was introduced by Pawlak as an uncertainty based model, which basically depends upon single equivalence relations defined over a universe or a set of equivalence relations, which are not considered simultaneously. Hence, from the granular computing point of view it is unigranular by nature. Qian et al in 2006 and in 2010 introduced two types of multigranular rough sets (MGRS) called the optimistic and pessimistic MGRS respectively. The stringent notion of mathematical equality of sets was extended by introducing a kind of approximate equality, called rough equality by Novotny and Pawlak, which uses basic rough sets. Later three more related types of such approximate equalities have been introduced by Tripathy et al. He has also provided a comparative analysis of these four types of approximate equalities of sets leading to approximate reasoning in real life situations. Two of these four types of approximate equalities; namely the rough equality and rough equivalence have been extended to the context of multigranulations by Tripathy et al very recently. In this paper we carry out this study further by introducing the notion of approximate rough equalities for multigranulations and establish their properties. We use a real life example to illustrate the results in the paper and also to construct examples in support of some parts of the properties.

The notion of rough sets introduced by Pawlak has been a successful model to capture impreciseness in data and has numerous applications. Since then it has been extended in several ways. The basic rough set introduced by Pawlak is a single granulation model from the granular computing point of view. Recently, this has been extended to two types of multigranular rough set models. Pawlak and Novotny introduced the notions of rough set equalities which is called approximate equalities. These notions of equalities use the user knowledge to decide the equality of sets and hence generate approximate reasoning. However, it was shown by Tripathy et al, even these notions have limited applicability to incorporate user knowledge. So the notion of rough equivalence was introduced by them. The notion of rough equalities in the multigranulation context was introduced and studied. In this article, we introduce the concepts of multigranular rough equivalences and establish their properties. Also, the replacement properties, which are obtained by interchanging the bottom equivalences with the top equivalences, have been established. We provide a real life example for both types of multigranulation, compare the rough multigranular equalities with the rough multigranular equivalences and illustrate the interpretation of the rough equivalences through the example.

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.

The notion of rough set captures indiscernibility of elements in a set. But, in many real life situations, an information system establishes the relation between different universes. This gave the extension of rough set on single universal set to rough set on two universal sets. In this paper, we introduce rough equality of sets on two universal sets and rough inclusion of sets employing the notion of the lower and upper approximation. Also, we establish some basic properties that refer to our knowledge about the universes.