Decomposition theorems and extension principle for complex Fuzzy sets

Document Type : Original Article


Faculty of Science and Engineering, University of Information Technology and Sciences, Vatara, Bangladesh.


Complex fuzzy set was originally proposed as a mathematical tool to deal with uncertainty by taking amplitude term and phase term memberships of an element of a universal set. In this article, we study (α,θ)-cut sets of the complex fuzzy sets and describe some related properties of them. Based on these (α,θ)-cut sets, some decomposition theorems of the complex fuzzy sets are proposed. Moreover, the concept of Zadeh’s extension principle of the fuzzy sets is extended to the complex fuzzy sets and explored various related properties. Finally, some arithmetic operations are demonstrated for the complex fuzzy set by using the extension principle of the complex fuzzy sets. Numerical illustrations for each arithmetic operation are also given.


Main Subjects

[1] L. Bi, B. Hu, S. Li, S. Dai, The Parallelity of complex fuzzy sets and parallelity preserving Operators, Journal of Intelligent & Fuzzy Systems, 34(6) 2018, 4173-4180.
[2] L. Bi, S. Dai, B. Hu, Complex fuzzy geometric aggregation Operators, Symmetry, 10(7) 2018, 251.
[3] L. Bi, S. Dai, B. Hu, S. Li, Complex fuzzy arithmetic aggregation operators, International Journal of Fuzzy System, 36(3) 2019, 2765-2771.
[4] B. Hu, L. Bi, S. Dai, Orthogonality between complex fuzzy sets and its application to signal detection, Symmetry, 9 (9) 2017, 175.
[5] B. Hu, L. Bi, S. Dai, S. Li, Approximate parallelity of complex fuzzy sets, Journal of Intelligent & Fuzzy Systems, 35(6) 2018, 6343-6351.
[6] B. Hu, L. Bi, S. Dai, S. Li, Distances of complex fuzzy sets and continuity of complex fuzzy operations, J Intell Fuzzy Syst., 35(2) 2018, 2247-2255.
[7] D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE Trans Fuzzy Syst., 10(2) 2002, 171-186.
[8] D. Ramot, M. Friedman, G. Langholz, A. Kandel, Complex fuzzy logic, IEEE Transaction on Fuzzy Systems, 11 2003, 450-461.
[9] P.K. Singh, Complex fuzzy concept lattice, Neural Process Lett. 2018.
[10] E.D. Tamir, J. Lin, A. Kandel, A new interpretation of complex membership grade, International Journal of Intelligent Systems, 26 2011, 285-312.
[11] P. Thirunavukarasu, R. Suresh, P. Thamilmani, Applications of Complex Fuzzy Sets, JP Journal of Applied Mathematics, 6(1-2) 2013, 5-22.
[12] L.A .Zadeh, Fuzzy sets, Inform. Control, 8 1965, 338-353.
[13] G. Zhang. T. S. Dillon, K. Y. Cai, J. Ma, J. Lu, δequalities of complex fuzzy relations, In Proceedings of the IEEE International 24th Conference on Advanced Information Networking and Applications, 2010, 1218-1224.
[14] G. Zhang. T.S. Dillon, K.Y. Cai, J. Ma, J. Lu, Operation properties and δ-equalities of complex fuzzy sets, International Journal of Approximate Reasoning, 50, 2009, 1227-1249.