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.
Similar content being viewed by others
References
Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249
Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Inf Control 31:312–340
Mizumoto M, Tanaka K (1981) Fuzzy sets of type 2 under algebraic product and algebraic sum. Fuzzy Sets Syst 31:277–290
Mendel JM, John RIB (2002) Type-2 fuzzy sets made simple. IEEE Trans Fuzzy Syst 10(2):117–127
Walker C, Walker E (2003) Algebraic Structures of fuzzy sets of type-2. In: Proceedings of International Conference on Fuzzy Information Processing March, 1-4:97–100
Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9(6):613–626
Dubois D, Prade H (1979) Operations in a fuzzy-valued Logic. Inf Control 43:224–240
Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Acdemic Press Inc., New York
Karnik NN, Mendel JM (1999) Applications of type-2 fuzzy logic systems to forecasting of time-series. Inf Sci 120:89–111
Karnik NN, Mendel JM (2001) Operations on type-2 fuzzy sets. Fuzzy Sets Syst 79:327–348
Wu D, Tan W (2006) A simplified type-2 fuzzy logic controller for real-time control. ISA Trans 45(4):503–516
Hu BQ, Wang CY (2014) On type-2 fuzzy relations and interval-valued type-2 fuzzy sets. Fuzzy Sets Syst 236:1–32
Mendel JM (2007) Computing with words and its relationship with fuzzistics. Inf Sci 177:988–1006
Gilan SS, Sebt MH, Shahhosseini V (2012) Computing with words for hierarchical competency based selection of personal in construction companies. Appl Soft Comput 12:860–871
Aliev R, Pedrycz W, Guirimov B, Aliev R, Iihan U, Babagil M, Mammadli S (2011) Type-2 fuzzy nueral networks with fuzzy clustering and differential evolution optimization. Inf Sci 181:1591–1608
Ozkan I, Turksen IB (2012) MiniMax e-star cluster validity index for type-2 fuzziness. Inf Sci 184:64–74
Choi B, Rhee F (2009) Interval type-2 fuzzy membership function generation methods for pattern recognition. Inf Sci 179:2102–2122
Dereli T, Baykasoglu A, Altun K, Durmusoglu A, Turksen B (2011) Industrial applications of type-2 fuzzy sets and systems: a concise review. Comput Ind 62:125–137
Leal-Ramirez C, Castillo O, Melin P, Rodriguez-Diaza A (2011) Simulation of the bird age-structured population growth based on an interval type-2 fuzzy cellular structure. Inf Sci 181:519–535
Chakravarty S, Dash PK (2012) A PSO based integrated functional link net and interval type-2 fuzzy logic system for predicting stock market indices. Appl Soft Comput 12:931–941
Tung SW, Quek C, Guan C (2013) eT2FIS: an evolving type-2 neural fuzzy inference system. Inf Sci 220:124–148
Kundu P, Kar S, Maiti M (2015) Multi-item solid transportation problem with type-2 fuzzy parameters. Appl Soft Comput 31:61–80
Murthy NVES, Lokavarapu S (2014) Lattice (Algebraic) properties of (Inverse) images of type-2 fuzzy subsets. Int J Sci Res 3(12):1741–1746
Acknowledgements
The work and research of the first author of this paper is financially supported by National Institute of Technology Silchar, Assam, India.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Statement of the Work in Broad Context
The theory of fuzzy relations is a generalization of crisp relations on a set. Fuzzy relations have been widely studied as a way to measure the degree of similarity between the objects of a given universe of discourse. Type-2 fuzzy relations further generalize the concept of type-1 fuzzy relations.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dutta, D., Sen, M. & Deshpande, A. Generalized Type-2 Fuzzy Equivalence Relation. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 92, 129–136 (2022). https://doi.org/10.1007/s40010-020-00707-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40010-020-00707-8