Algebraic Properties of Rough Set on Two Universal Sets based on Multigranulation

Introduction

The traditional concept of crisp sets has been extended in many directions as far as modeling of real life situations is concerned. The earliest is the notion of fuzzy set by l. A. Zadeh (1965) that captures impreciseness in information. On the other hand rough sets of z. Pawlak (1982, 1991) capture indiscernibility among objects to model imperfect knowledge. The basic philosophy is that human knowledge about a universe depends upon their capability to classify its objects. So, classification of a universe and indiscernibility relations defined on it are known to be interchangeable notions. The basic idea of rough set is based upon the approximation of sets by pair of sets known as lower approximation and upper approximation. Here, the lower and upper approximation operators are based on equivalence relations. However, the requirement of equivalent relation is a restrictive one and failure in many real life situations. In order to achieve this, rough set is generalized to binary relations (yao, (1998,2001,2004); kondo, 2006; pawlak & skowron, 2007a), fuzzy proximity relations (tripathy & acharjya (2008, 2010)), intuitionistic fuzzy proximity relations (tripathy, 2006; tripathy & acharjya (2009, 2011)), boolean algebras (liu, 2005; pawlak & skowron, 2007b), fuzzy lattices (liu, 2008), completely distributive lattices (chen et. Al., 2006) and neighborhood systems (lin, 1989). Development of these techniques and tools is studied under different domains like knowledge discovery in database, computational intelligence, knowledge representation, granular computing etc. (saleem durai et al., 2012; acharjya et al. (2011, 2012); tripathy et al., 2011).

In the view of granular computing, a general concept described by a set is always characterized by lower and upper approximations under static granulation. It indicates that the concept is depicted by means of single equivalence relation on the universe. However, in many real life situations, many concepts are described by using multi equivalence relations. Therefore, basic rough set model has been extended to rough set on multigranulations (qian et al., (2006, 2007)) in which the set approximations are defined by using multi-equivalences on the universe. On the other hand, rough set models on two universal sets are generalized with generalized approximation spaces and interval structure (wong et al., 1993). Here, the equivalence relation is generalized to binary relation.