Document Type : Research Paper


1 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India.

2 Department of Mathematics, Chaudhary Charan Singh University, Meerut, India-250004.

3 Ward Number – 16, Bhagatbandh, Anuppur 484 224, Madhya Pradesh, India.


This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG)  in the context of -norm and  -conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the -norm for "min" and the -conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the -norm and -conorm, and discuss important properties associated with them.


Main Subjects

Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. DOI:10.1016/S0019-9958(65)90241-X
Rosenfeld, A. (1971). Fuzzy groups. Journal of mathematical analysis and applications, 35(3), 512–517. DOI:10.1016/0022-247X(71)90199-5
Anthony, J. M., & Sherwood, H. (1979). Fuzzy groups redefined. Journal of mathematical analysis and applications, 69(1), 124–130. DOI:10.1016/0022-247X(79)90182-3
Bhattacharya, P., & Mukherjee, N. P. (1985). Fuzzy relations and fuzzy groups. Information sciences, 36(3), 267–282. DOI:10.1016/0020-0255(85)90057-X
Atanassov, A. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87-96. DOI:10.5555/1708507.1708520
Das, P. S. (1981). Fuzzy groups and level subgroups. Journal of mathematical analysis and applications, 84(1), 264–269. DOI:10.1016/0022-247X(81)90164-5
Ajmal, N., & Prajapati, A. S. (1992). Fuzzy cosets and fuzzy normal subgroups. Information sciences, 64(1–2), 17–25. DOI:10.1016/0020-0255(92)90107-J
Ajmal, N., & Thomas, K. V. (1994). The lattices of fuzzy subgroups and fuzzy normal subgroups. Information sciences, 76(1–2), 1–11. DOI:10.1016/0020-0255(94)90064-7
Dixit, V. N., Kumar, R., & Ajmal, N. (1990). Level subgroups and union of fuzzy subgroups. Fuzzy sets and systems, 37(3), 359–371. DOI:10.1016/0165-0114(90)90032-2
Gau, W. L., & Buehrer, D. J. (1993). Vague Sets. IEEE transactions on systems, man and cybernetics, 23(2), 610–614. DOI:10.1109/21.229476
Khan, H., Ahmad, M., & Biswas, R. (2007). On vague groups. International journal of computational cognition, 5(1), 27–30.
Yager, R. R. (2013). Pythagorean fuzzy subsets [presentation]. Proceedings of the 2013 joint ifsa world congress and nafips annual meeting, ifsa/nafips 2013 (pp. 57–61). DOI: 10.1109/IFSA-NAFIPS.2013.6608375
Rasuli, R. (2017). Fuzzy subgroups on direct product of groups over a $t$-norm. Journal of fuzzy set valued analysis, 2017(3), 96–101. DOI:10.5899/2017/jfsva-00339
Rasuli, R. (2019). Fuzzy equivalence relation, fuzzy congrunce relation and fuzzy normal subgroups on group G over T-Norms. Asian journal of fuzzy and applied mathematics, 7(2). DOI:10.24203/ajfam.v7i2.5736
Rasuli, R. (2020). Intuitionistic fuzzy subgroups with respect to norms ( T , S ). Engineering and applied science letters, 3(2), 40–53. DOI:10.30538/psrp-easl2020.0040
Gayen, S., Smarandache, F., Jha, S., & Kumar, R. (2020). Interval-valued neutrosophic subgroup based on interval-valued triple T-Norm. In Neutrosophic sets in decision analysis and operations research (pp. 215–243). IGI Global. DOI: 10.4018/978-1-7998-2555-5.ch010
Senapati, T., & Yager, R. R. (2020). Fermatean fuzzy sets. Journal of ambient intelligence and humanized computing, 11(2), 663–674. DOI:10.1007/s12652-019-01377-0
Bejines, C., Chasco, M. J., & Elorza, J. (2021). Aggregation of fuzzy subgroups. Fuzzy sets and systems, 418, 170–184. DOI:10.1016/j.fss.2020.05.017
Ardanza-Trevijano, S., Chasco, M. J., Elorza, J., de Natividade, M., & Talavera, F. J. (2023). Aggregation of T-subgroups. Fuzzy sets and systems, 463, 108390. DOI:10.1016/j.fss.2022.08.022
Kumar, T., Dhiman, N., Vandana, Vashistha, S., Sharma, M. K., & Mishra, V. N. (2020). Vague groups redefined with respect to t-norm. Advances in mathematics: scientific journal, 9(10), 8475–8483. DOI:10.37418/amsj.9.10.76
Bhunia, S., Ghorai, G., & Xin, Q. (2021). On the characterization of pythagorean fuzzy subgroups. AIMS mathematics, 6(1), 962–978. DOI:10.3934/math.2021058
Razaq, A., Alhamzi, G., Razzaque, A., & Garg, H. (2022). A comprehensive study on Pythagorean fuzzy normal subgroups and Pythagorean fuzzy isomorphisms. Symmetry, 14(10), 2084. DOI:10.3390/sym14102084
Boixader, D., & Recasens, J. (2022). Vague and fuzzy t-norms and t-conorms. Fuzzy sets and systems, 433, 156–175. DOI:10.1016/j.fss.2021.07.008
Silambarasan, I. (2021). Fermatean fuzzy subgroups. Journal of the international mathematical virtual institute, 11(1), 1–16.