Algebraic structure of some complex intuitionistic fuzzy subgroups and their homomorphism

Zhuonan Wu; Zengtai Gong; Zhuonan Wu; Zengtai Gong

doi:10.3934/math.2025189

AIMS Mathematics

2025, Volume 10, Issue 2: 4067-4091. doi: 10.3934/math.2025189

Previous Article Next Article

Research article Topical Sections

Algebraic structure of some complex intuitionistic fuzzy subgroups and their homomorphism

Zhuonan Wu ,
Zengtai Gong ^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 26 December 2024 Revised: 13 February 2025 Accepted: 19 February 2025 Published: 27 February 2025
MSC : 03E72, 08A72, 26E50

The complex fuzzy environment is an innovative tool to deal with fuzzy situations in different mathematical problems. Aiming at the concept of complex intuitionistic fuzzy subgroups, this paper has introduced cut-subsets of complex intuitionistic fuzzy sets, and studied the relationship among the cut-subsets and complex intuitionistic fuzzy subgroups, complex intuitionistic fuzzy Abel subgroups, and complex intuitionistic fuzzy cyclic subgroups. Further, we gave the left and right cosets of complex intuitionistic fuzzy subgroups, defined complex intuitionistic fuzzy normal subgroups, and discussed some of their algebraic properties. Based on this thought, we proposed a new concept of $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy subgroups, and proved that an $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. At the same time, $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy normal subgroups and their cosets were introduced. Finally, we established a general homomorphism of complex intuitionistic fuzzy subgroups, and studied the relationship between the image and pre-image of complex intuitionistic fuzzy subgroups and complex intuitionistic fuzzy normal subgroups, respectively, under group homomorphism.
Citation: Zhuonan Wu, Zengtai Gong. Algebraic structure of some complex intuitionistic fuzzy subgroups and their homomorphism[J]. AIMS Mathematics, 2025, 10(2): 4067-4091. doi: 10.3934/math.2025189

Related Papers:

Abstract

The complex fuzzy environment is an innovative tool to deal with fuzzy situations in different mathematical problems. Aiming at the concept of complex intuitionistic fuzzy subgroups, this paper has introduced cut-subsets of complex intuitionistic fuzzy sets, and studied the relationship among the cut-subsets and complex intuitionistic fuzzy subgroups, complex intuitionistic fuzzy Abel subgroups, and complex intuitionistic fuzzy cyclic subgroups. Further, we gave the left and right cosets of complex intuitionistic fuzzy subgroups, defined complex intuitionistic fuzzy normal subgroups, and discussed some of their algebraic properties. Based on this thought, we proposed a new concept of $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy subgroups, and proved that an $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy subgroup is a general form of every complex intuitionistic fuzzy subgroup. At the same time, $ (\alpha_{1, 2}, \beta_{1, 2}) $-complex intuitionistic fuzzy normal subgroups and their cosets were introduced. Finally, we established a general homomorphism of complex intuitionistic fuzzy subgroups, and studied the relationship between the image and pre-image of complex intuitionistic fuzzy subgroups and complex intuitionistic fuzzy normal subgroups, respectively, under group homomorphism.

References

[1]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
[2]	A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl, 35 (1971), 512–517. https://doi.org/10.1016/0022-247X(71)90199-5 doi: 10.1016/0022-247X(71)90199-5
[3]	K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[4]	R. Biswas, Intuitionistic fuzzy subgroups, Mathematical Forum, 10 (1989), 37–46.
[5]	P. K. Sharma, Homomorphism of intuitionistic fuzzy groups, International Mathematical Forum, 6 (2011), 3169–3178.
[6]	P. K. Sharma, Intuitionistic fuzzy groups, Int. J. Data Warehous., 1 (2011), 86–94.
[7]	M. F. Marashdeh, A. R. Salleh, Intuitionistic fuzzy groups, Asian Journal of Algebra, 2 (2009), 1–10. https://doi.org/10.3923/aja.2009.1.10 doi: 10.3923/aja.2009.1.10
[8]	M. F. Marashdeh, A. R. Salleh, The intuitionistic fuzzy normal subgroup, Int. J. Fuzzy Log. Inte., 10 (2010), 82–88. https://doi.org/10.5391/IJFIS.2010.10.1.082 doi: 10.5391/IJFIS.2010.10.1.082
[9]	P. K. Sharma, t-Intuitionistic fuzzy subgroups, International Journal of Fuzzy Mathematics and Systems, 2 (2012), 233–243.
[10]	M. Gulzar, D. Alghazzawi, M. H. Mateen, N. Kausar, A certain class of t-intuitionistic fuzzy subgroups, IEEE Access, 8 (2020), 163260–163268. https://doi.org/10.1109/ACCESS.2020.3020366 doi: 10.1109/ACCESS.2020.3020366
[11]	D. Ramot, R. Milo, M. Friedman, A. Kandel, Complex fuzzy sets, IEEE T. Fuzzy Syst., 10 (2002), 171–186. https://doi.org/10.1109/91.995119 doi: 10.1109/91.995119
[12]	A. S. Alkouri, A. R. Salleh, Complex intuitionistic fuzzy sets, AIP Conf. Proc., 1482 (2012), 464–470. https://doi.org/10.1063/1.4757515 doi: 10.1063/1.4757515
[13]	A. Al-Husban, A. R. Salleh, Complex fuzzy group based on complex fuzzy space, Global Journal of Pure and Applied Mathematics, 12 (2016), 1433–1450.
[14]	M. O. Alsarahead, A. G. Ahmad, Complex fuzzy soft subgroups, J. Qual. Meas. Anal., 13 (2017), 17–28.
[15]	M. O. Alsarahead, A. G. Ahmad, Complex fuzzy subgroups, Appl. Math. Sci., 11 (2017), 2011–2021. https://doi.org/10.12988/ams.2017.64115 doi: 10.12988/ams.2017.64115
[16]	M. O. Alsarahead, A. G. Ahmad, Complex fuzzy subrings, International Journal of Pure and Applied Mathematics, 117 (2017), 563–577. https://doi.org/10.12732/ijpam.v117i4.1 doi: 10.12732/ijpam.v117i4.1
[17]	W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Set. Syst., 8 (1982), 133–139. https://doi.org/10.1016/0165-0114(82)90003-3 doi: 10.1016/0165-0114(82)90003-3
[18]	H. Alolaiyan, H. A. Alshehri, M. H. Mateen, D. Pamucar, M. Gulzar, A novel algebraic structure of $(\alpha, \beta)$-complex fuzzy subgroups, Entropy, 23 (2021), 992. https://doi.org/10.3390/e23080992 doi: 10.3390/e23080992
[19]	D. Al-Sharoa, $(\alpha_{1, 2}, \beta_{1, 2})$-Complex intuitionistic fuzzy subgroups and its algebraic structure, AIMS Mathematics, 8 (2023), 8082–8116. https://doi.org/10.3934/math.2023409 doi: 10.3934/math.2023409
[20]	R. Al-Husbana, A. R. Salleh, A. G. Ahmad, Complex intuitionistic fuzzy group subrings, AIP Conf. Proc., 1784 (2016), 050006. https://doi.org/10.1063/1.4966825 doi: 10.1063/1.4966825
[21]	R. Al-Husbana, A. R. Salleh, A. G. Ahmad, Complex intuitionistic fuzzy normal subgroup, International Journal of Pure and Applied Mathematics, 115 (2017), 455–466. https://doi.org/10.12732/ijpam.v115i3.1 doi: 10.12732/ijpam.v115i3.1
[22]	F. Y. Xiao, Generalized belief function in complex evidence theory, J. Intell. Fuzzy Syst., 38 (2020), 3665–3673. https://doi.org/10.3233/JIFS-179589 doi: 10.3233/JIFS-179589
[23]	F. Y. Xiao, Quantum X-entropy in generalized quantum evidence theory, Inform. Sciences, 643 (2023), 119177. https://doi.org/10.1016/j.ins.2023.119177 doi: 10.1016/j.ins.2023.119177
[24]	F. Y. Xiao, W. Pedrycz, Negation of the quantum mass function for multisource quantum information fusion with its application to pattern classification, IEEE T. Pattern Anal., 45 (2023), 2054–2070. https://doi.org/10.1109/TPAMI.2022.3167045 doi: 10.1109/TPAMI.2022.3167045
[25]	J. Rezaei, Best-worst multi-criteria decision-making method, Oemga, 53 (2015), 49–57. https://doi.org/10.1016/j.omega.2014.11.009 doi: 10.1016/j.omega.2014.11.009
[26]	S. P. Wan, J. Y. Dong, A novel extension of best-worst method with intuitionistic fuzzy reference comparisons, IEEE T. Fuzzy Syst., 30 (2022), 1698–1711. https://doi.org/10.1109/TFUZZ.2021.3064695 doi: 10.1109/TFUZZ.2021.3064695
[27]	J. Y. Dong, S. P. Wang, Interval-valued intuitionistic fuzzy best-worst method with additive consistency, Expert Syst. Appl., 236 (2024), 121213. https://doi.org/10.1016/j.eswa.2023.121213 doi: 10.1016/j.eswa.2023.121213
[28]	S. P. Wan, J. Y. Dong, S. M. Chen, A novel intuitionistic fuzzy best-worst method for group decision making with intuitionistic fuzzy preference relations, Inform. Sciences, 666 (2024), 120404. https://doi.org/10.1016/j.ins.2024.120404 doi: 10.1016/j.ins.2024.120404
[29]	X. Y. Lu, J. Y. Dong, S. P. Wan, H. C. Li, Interactively iterative group decision-making method with interval-valued intuitionistic fuzzy preference relations based on a new additively consistent concept, Appl. Soft Comput., 152 (2024), 111199. https://doi.org/10.1016/j.asoc.2023.111199 doi: 10.1016/j.asoc.2023.111199
[30]	S. P. Wan, S. Z. Gao, J. Y. Dong, Trapezoidal cloud based heterogeneous multi-criterion group decision-making for container multimodal transport path selection, Appl. Soft Comput., 154 (2024), 111374. https://doi.org/10.1016/j.asoc.2024.111374 doi: 10.1016/j.asoc.2024.111374
[31]	S. P. Wan, W. C. Zou, J. Y. Dong, Y. Gao, Dual strategies consensus reaching process for ranking consensus based probabilistic linguistic multi-criteria group decision-making methos, Expert Syst. Appl., 262 (2025), 125342. https://doi.org/10.1016/j.eswa.2024.125342 doi: 10.1016/j.eswa.2024.125342
[32]	H. R. Zhang, Foundation of modern algebra, Beijing: Higher Education Press, 2011.
[33]	M. Gulzar, M. H. Mateen, D. Alghazzawi, N. Kausar, A novel applications of complex intuitionistic fuzzy sets in group theorem, IEEE Access, 8 (2020), 196075–196085. https://doi.org/10.1109/ACCESS.2020.3034626 doi: 10.1109/ACCESS.2020.3034626

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)