Picture fuzzy multifunctions and modal topological structures

M. N. Abu_Shugair; A. A. Abdallah; Malek Alzoubi; S. E. Abbas; Ismail Ibedou; M. N. Abu_Shugair; A. A. Abdallah; Malek Alzoubi; S. E. Abbas; Ismail Ibedou

doi:10.3934/math.2025341

AIMS Mathematics

2025, Volume 10, Issue 3: 7430-7448. doi: 10.3934/math.2025341

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Research article Topical Sections

Picture fuzzy multifunctions and modal topological structures

1.
Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia
2.
Department of Computer science, College of engineering and computer sciences, Jazan University, Jazan 45142, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
4.
Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

Received: 06 January 2025 Revised: 20 March 2025 Accepted: 25 March 2025 Published: 31 March 2025
MSC : 03B20, 03B52, 03E72, 03E75, 94D05

This paper introduces the notion of picture fuzzy modal topological multifunctions. These structures are grounded on novel picture fuzzy topological operators for closure and interior types, thereby utilizing the two standard picture fuzzy modal operators $ \square $ and $ \Diamond $. The paper discusses several fundamental properties of picture fuzzy multifunctions. Many types of structures are introduced, and some types of continuous multifunctions between picture fuzzy topological structures are discussed. The results indicate that some properties considered satisfactory in the intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2022, are not fulfilled.
- picture fuzzy multifunction,
- picture fuzzy modal topology,
- picture fuzzy operator
Citation: M. N. Abu_Shugair, A. A. Abdallah, Malek Alzoubi, S. E. Abbas, Ismail Ibedou. Picture fuzzy multifunctions and modal topological structures[J]. AIMS Mathematics, 2025, 10(3): 7430-7448. doi: 10.3934/math.2025341

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Abstract

This paper introduces the notion of picture fuzzy modal topological multifunctions. These structures are grounded on novel picture fuzzy topological operators for closure and interior types, thereby utilizing the two standard picture fuzzy modal operators $ \square $ and $ \Diamond $. The paper discusses several fundamental properties of picture fuzzy multifunctions. Many types of structures are introduced, and some types of continuous multifunctions between picture fuzzy topological structures are discussed. The results indicate that some properties considered satisfactory in the intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2022, are not fulfilled.

References

[1]	A. Razaq, I. Masmali, H. Garg, U. Shuaib, Picture fuzzy topological spaces and associated continuous functions, AIMS Mathematics, 7 (2022), 14840–14861. http://dx.doi.org/10.3934/math.2022814 doi: 10.3934/math.2022814
[2]	M. Abu_Shugair, A. Abdallah, S. Abbas, I. Ibedou, Double fuzzy $\alpha $-$\eth$-continuous multifunctions, AIMS Mathematics, 9 (2024), 16623–16642. http://dx.doi.org/10.3934/math.2024806 doi: 10.3934/math.2024806
[3]	M. Abu_Shugair, A. Abdallah, S. Abbas, E. El-Sanowsy, I. Ibedou, Double fuzzy ideal multifunctions, Mathematics, 12 (2024), 1128. http://dx.doi.org/10.3390/math12081128 doi: 10.3390/math12081128
[4]	T. Al-shami, A. Mhemdi, Generalized frame for orthopair fuzzy sets: $(m, n)$-fuzzy sets and their applications to multi-criteria decision-making methods, Information, 14 (2023), 56. http://dx.doi.org/10.3390/info14010056 doi: 10.3390/info14010056
[5]	I. Alshammari, P. Mani, C. Ozel, H. Garg, Multiple attribute decision making algorithm via picture fuzzy nano topological spaces, Symmetry, 13 (2021), 69. http://dx.doi.org/10.3390/sym13010069 doi: 10.3390/sym13010069
[6]	S. Ashraf, S. Abdullah, T. Mahmood, F. Ghani, T. Mahmood, Spherical fuzzy sets and their applications in multi-attribute decision making problems, J. Intell. Fuzzy Syst., 36 (2019), 2829–2844. http://dx.doi.org/10.3233/JIFS-172009 doi: 10.3233/JIFS-172009
[7]	K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
[8]	K. Atanassov, Intuitionistic fuzzy modal topological structure, Mathematics, 10 (2022), 3313. http://dx.doi.org/10.3390/math10183313 doi: 10.3390/math10183313
[9]	K. Atanassov, R. Tsvetkov, New intuitionistic fuzzy operations, operators and topological structures, Iran. J. Fuzzy Syst., 20 (2023), 37–53. http://dx.doi.org/10.22111/IJFS.2023.7629 doi: 10.22111/IJFS.2023.7629
[10]	K. Atanassov, N. Angelova, T. Pencheva, On two intuitionistic fuzzy modal topological structures, Axioms, 12 (2023), 408. http://dx.doi.org/10.3390/axioms12050408 doi: 10.3390/axioms12050408
[11]	P. Chellamani, D. Ajay, S. Broumi, T. Ligori, An approach to decision-making via picture fuzzy soft graphs, Granul. Comput., 7 (2022), 527–548. http://dx.doi.org/10.1007/s41066-021-00282-2 doi: 10.1007/s41066-021-00282-2
[12]	B. Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernetics, 30 (2014), 409. http://dx.doi.org/10.15625/1813-9663/30/4/5032 doi: 10.15625/1813-9663/30/4/5032
[13]	M. Fitting, R. Mendelsohn, First-order modal logic, Cham: Springer, 2023. http://dx.doi.org/10.1007/978-3-031-40714-7
[14]	H. Garg, M. Atef, $Cq$-ROFRS: covering $q$-rung orthopair fuzzy rough sets and its application to multi-attribute decision making process, Complex Intell. Syst., 8 (2022), 2349–2370. http://dx.doi.org/10.1007/s40747-021-00622-4 doi: 10.1007/s40747-021-00622-4
[15]	R. Joshi, S. Kumar, A novel VIKOR approach based on weighted correlation coefficients and picture fuzzy information for multicriteria decision making, Granul. Comput., 7 (2022), 323–336. http://dx.doi.org/10.1007/s41066-021-00267-1 doi: 10.1007/s41066-021-00267-1
[16]	F. Gündoĝdu, C. Kahraman, Spherical fuzzy sets and spherical fuzzy TOPSIS method, J. Intell. Fuzzy Syst., 36 (2019), 337–352. http://dx.doi.org/10.3233/JIFS-181401 doi: 10.3233/JIFS-181401
[17]	L. Li, R. Zhang, J. Wang, X. Shang, K. Bai, A novel approach to multi-attribute group decision-making with $q$-rung picture linguistic information, Symmetry, 10 (2018), 172. http://dx.doi.org/10.3390/sym10050172 doi: 10.3390/sym10050172
[18]	M. Luo, Y. Zhang, A new similarity measure between picture fuzzy sets and its application, Eng. Appl. Artif. Intel., 96 (2020), 103956. http://dx.doi.org/10.1016/j.engappai.2020.103956 doi: 10.1016/j.engappai.2020.103956
[19]	G. Mints, A short introduction to modal logic, Chicago: University of Chicago Press, 1992.
[20]	M. Olgun, M. Ünver, S. Yardmc, Pythagorean fuzzy points and applications in pattern recognition and Pythagorean fuzzy topologies, Soft Comput., 25 (2021), 5225–5232. http://dx.doi.org/10.1007/s00500-020-05522-2 doi: 10.1007/s00500-020-05522-2
[21]	T. Senapati, R. Yager, Fermatean fuzzy weighted averaging/geometric operators and its application in multi-criteria decision making methods, Eng. Appl. Artif. Intell., 85 (2019), 112–121. http://dx.doi.org/10.1016/j.engappai.2019.05.012 doi: 10.1016/j.engappai.2019.05.012
[22]	G. Wei, Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making, Informatica, 28 (2017), 547–564. http://dx.doi.org/10.3233/INF-2017-1150 doi: 10.3233/INF-2017-1150
[23]	R. Yager, Pythagorean fuzzy subsets, Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), 2013, 57–61. http://dx.doi.org/10.1109/IFSA-AFIPS.2013.6608375 doi: 10.1109/IFSA-AFIPS.2013.6608375
[24]	R. Yager, Generalized orthopair fuzzy sets, IEEE Trans. Fuzzy Syst., 25 (2017), 1222–1230. http://dx.doi.org/10.1109/TFUZZ.2016.2604005 doi: 10.1109/TFUZZ.2016.2604005
[25]	Y. Yang, C. Liang, S. Ji, T. Liu, Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making, J. Intell. Fuzzy Syst., 29 (2015), 1711–1722. http://dx.doi.org/10.3233/IFS-151648 doi: 10.3233/IFS-151648
[26]	J. Ye, Similarity measures based on the generalized distance of neutrosophic $Z$-number sets and their multi-attribute decision making method, Soft Comput., 25 (2021), 13975–13985. http://dx.doi.org/10.1007/s00500-021-06199-x doi: 10.1007/s00500-021-06199-x
[27]	A. Yolcu, F. Smarandache, T. Öztürk, Intuitionistic fuzzy hypersoft sets, Commun. Fac. Sci. Univ., 70 (2021), 443–455. http://dx.doi.org/10.31801/cfsuasmas.788329 doi: 10.31801/cfsuasmas.788329
[28]	L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X

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