On Multi-Fuzzy Rough Sets, Relations, and Topology

On Multi-Fuzzy Rough Sets, Relations, and Topology

Gayathri Varma (Department of Mathematics, National Institute of Technology, Kerala, India) and Sunil Jacob John (Department of Mathematics, National Institute of Technology, Kerala, India)
Copyright: © 2019 |Pages: 19
DOI: 10.4018/IJFSA.2019010106


This article describes how rough set theory has an innate topological structure characterized by the partitions. The approximation operators in rough set theory can be viewed as the topological operators namely interior and closure operators. Thus, topology plays a role in the theory of rough sets. This article makes an effort towards considering closed sets a primitive concept in defining multi-fuzzy topological spaces. It discusses the characterization of multi-fuzzy topology using closed multi-fuzzy sets. A set of axioms is proposed that characterizes the closure and interior of multi-fuzzy sets. It is proved that the set of all lower approximation of multi-fuzzy sets under a reflexive and transitive multi-fuzzy relation forms a multi-fuzzy topology.
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2. Literature Survey

The chief aspect of fuzzy sets is that their boundaries are not precise. There exists an alternative way to formulate sets with imprecise boundaries. Sets formulated in this way are called rough sets. Zdsizlaw Pawlak (1982) laid the foundation of rough set theory. A rough set is basically an approximate representation of a given crisp set in terms of two subsets of a crisp partition defined on the universal set involved. The two subsets are called a lower approximation and an upper approximation. Rough set theory, has far reaching applications in numerous fields like data mining, machine learning, knowledge discovery, pattern recognition etc. The theory of rough sets and its various applications are well covered in (Pawlak, 2012) and (Polkowski, 2013).

Rough set theory and fuzzy set theory are not rivaling theories but rather complement each other. Both are different and independent approach to imperfect knowledge. Scholars like Dubois and Prade (1990) carried out studies on combining fuzzy set theory and rough set theory and obtained hybrid structures namely rough fuzzy sets and fuzzy rough sets. Guided by this observation, rough set theory is integrated with other fuzzy set theories so that more general structures like interval valued fuzzy rough sets (Zhang, Zhang, & Wu, 2009; Zhang, 2013) intuitionistic fuzzy rough sets (Zhou, Wu, & Zhang, 2009; Zhou & Wu, 2008), hesitant fuzzy rough sets (Deepak & John, 2014) multi-fuzzy rough sets (Varma & John, 2014) etc. are obtained. These investigations have shown how soft technology theories can be combined fruitfully to attain a more flexible and expressive framework that helps in modelling incomplete information in the available knowledge base.

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