Generalized Type-2 Fuzzy Equivalence Relation
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RESEARCH ARTICLE
Generalized Type-2 Fuzzy Equivalence Relation Dhiman Dutta1 • Mausumi Sen1 • Ashok Deshpande2
Received: 4 October 2019 / Revised: 30 July 2020 / Accepted: 5 August 2020 Ó The National Academy of Sciences, India 2020
Abstract This paper extends the concept of generalized equivalence relation on type-2 fuzzy set and presents a comprehensive study of type-2 fuzzy G-equivalence relation. Notions like partition of a type-2 fuzzy G-equivalence relation, type-2 fuzzy balance mappings in the type-2 fuzzy set theory are introduced and related theorems are discussed. Keywords Type-2 fuzzy G-equivalence relation Partition of a type-2 fuzzy G-equivalence relation Type-2 fuzzy balanced mappings
1 Introduction The notion of type-2 fuzzy set was defined by Zadeh [1] as a generalization of type-1 fuzzy set. The membership function of a type-1 fuzzy set is a real number in [0, 1], while the membership function of a type-2 fuzzy set is a type-1 fuzzy set. However, type-2 fuzzy sets did not get much attention from researchers because it was difficult to understand and develop link with the type-1 fuzzy sets. Lately, several papers have been published on type-2 fuzzy & Dhiman Dutta [email protected] Mausumi Sen [email protected] Ashok Deshpande [email protected] 1
Department of Mathematics, National Institute of Technology Silchar, Assam, India
2
Berkeley Initiative in Soft Computing, University of California, Berkeley, CA, USA
sets to extend the classical mathematical concepts and the type-1 fuzzy sets mathematical concepts to the case of type-2 fuzzy sets. Mizumoto and Tanaka [2, 3] studied properties of membership grades of type-2 fuzzy sets, settheoretic operations of such sets. Mendel and John [4] investigated the operations for union, intersection and complement of type-2 fuzzy sets with superior acceptance deprived of the requirement of operating the difficult extension principle. Walker and Walker [5] investigated the algebraic operations in the type-2 fuzzy sets. More algebraic studies on type-2 fuzzy sets had been shown by Dubois and Prade [6–8], Karnik and Mendel [9, 10] and Wu and Tan [11]. The concept of cartesian product of type2 fuzzy sets was given by Hu and Yang [12] as an extension of type-1 fuzzy sets. Dubois and Prade [6–8] discussed the composition of type-2 fuzzy relations and presented a formulation only for minimum t-norm which is, perhaps, an extension of type-1 sup-star composition. Type-2 fuzzy sets and fuzzy inference system is being applied in various areas of science and technology such as: computing with words [13], human resource management [14], forecasting of time-series [9], clustering [15, 16], pattern recognition [17], fuzzy logic controller [11], industrial application [18], simulation [19], neural network [20, 21], and solid transportation problem [22]. Notions like type-2 fuzzy G-reflexive relations are interesting generalizations of the concept of classical reflexivity of type-2 fuzzy sets. It can be noticed that ~ xÞ ¼ 1 seems to be strong to
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