Picture Fuzzy Lattices, Ideals and Homomorphism
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Mathematics and Statistics, Bangladesh University of Business and Technology, Dhaka, Bangladesh.
Keywords: Picture fuzzy set, Picture fuzzy lattice, Picture fuzzy ideal, Picture fuzzy homomorphism, Picture fuzzy epimomorphism.,
Abstract :
Picture fuzzy set (PFS) is a novel concept for dealing with uncertainty and a generalization of the traditional fuzzy set (FS) and intuitionistic fuzzy set (IFS) and can easily manage the uncertain nature of human thoughts by incorporating the positive, neutral, negative and refusal membership degrees of an object. Many conceptual ideas on PFSs have been developed so far and applied in diversified fields. In this paper, the concept of picture fuzzy sublattices and picture fuzzy ideals are developed and some of their associated properties are established in detail. Moreover, the sum and product of two picture fuzzy ideals are introduced with their properties. Finally, some properties of picture fuzzy ideals under lattice homomorphism are explored.
[1] Zadeh LA. Fuzzy sets. Information and Control. 1965; 8(3): 338-356.
[2] Atanassov KT. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20(1): 87-96.
[3] Cuong BC, Kreinovich V. Picture fuzzy setsa new concept for computational intelligence problems. In: Proceedings of the Third World Congress on Information and Communication Technologies (WICT). 2013. pp. 16. DOI: https://doi.org/10.1109/WICT.2013.7113099
[4] Cuong BC. Picture fuzzy sets. Journal of Computer Science and Cybernetics. 2014; 30(4): 409-420.
[5] Hasan MK, Ali MY, Sultana A, Mitra NK. Extension principles for picture fuzzy sets. Journal of Intelligent & Fuzzy Systems. 2023; 44: 6265-6275. DOI: https://doi.org/10.3233/JIFS-220616
[6] Hasan MK, Sultana A, Mitra NK. Arithmetic operations of generalized trapezoidal picture fuzzy numbers by vertex method. American Journal of Computational Mathematics. 2023; 13(1): 99-121. DOI: https://doi.org/10.4236/ajcm.2023.131004
[7] Hasan MK, Sultana A, Mitra NK. Picture fuzzy relations over picture fuzzy sets. American Journal of Computational Mathematics. 2023; 13(1): 161-184. DOI: https://doi.org/10.4236/ajcm.2023.131008
[8] Hasan MK, Sultana A, Mitra NK. Compositions of picture fuzzy relations with application in decision making. European Journal of Mathematics and Statistics. 2023; 4(2): 19-28. DOI: https://doi.org/10.24018/ejmath.2023.4.2.188
[9] Hasan MK, Sultana A, Mitra NK. Arithmetic operations of generalized triangular picture fuzzy numbers with applications. Fuzzy Optimization and Modeling Journal. 2023; 4(4): 49-69. DOI: https://doi.org/10.30495/fomj.2023.1994693.1110
[10] Hasan MK, Ali MY, Sultana A, Mitra NK. Minimal decomposition theorems and minimal extension principle for picture fuzzy sets. Journal of Mathematics and Computer Science. 2022; 12: 120.
[11] Hasan MK, Ali MY, Sultana A, Mitra NK. Some picture fuzzy mean operators and their applications in decision-making. Journal of Fuzzy Extension and Applications. 2022; 3(4): 349-361. DOI: https://doi.org/10.22105/jfea.2022.341302.1219
[12] Ali MY, Khan IA, Hasan MK. Tangent similarity measures of picture fuzzy sets and their applications to pattern recognition. UITS Journal of Science & Engineering. 2023; 8-9(1): 55-62.
[13] Rosenfeld A. Fuzzy groups. Journal of Mathematical Analysis and Applications. 1971; 35: 512-517.
[14] Dixit VN, Kumar R, Ajmal N. Level subgroups and union of fuzzy subgroups. Fuzzy Sets and Systems. 1990; 37: 359-371. DOI: https://doi.org/10.1016/0165-0114(90)90032-2
[15] Eroglu MS. The homomorphic image of a fuzzy subgroup is always a fuzzy subgroup. Fuzzy Sets and Systems. 1989; 53: 255-256. DOI: https://doi.org/10.1016/0165-0114(89)90246-7
[16] Liu WJ. Fuzzy invariant subgroups and fuzzy ideals. Fuzzy Sets and Systems. 1982; 21: 133-139. DOI: https://doi.org/10.1016/0165-0114(82)90003-3
[17] Mordeson JN, Malik DS. Fuzzy Commutative Algebra. USA: World Scientific Publishing; 1998.
[18] Swamy UM, Raju V. Fuzzy ideals and congruence of lattices. Fuzzy Sets and Systems. 1998; 94-103. DOI: https://doi.org/10.1016/S0165-0114(96)00310-7
[19] Ajmal N, Thomas KV. Homomorphism of fuzzy subgroups, correspondences theorem and fuzzy quotient group. Fuzzy Sets and Systems. 1994; 61(3): 329-339.
[20] Bharathi P, Vimala J. The role of fuzzy l-ideals in a commutative lattice ordered group. Global Journal of Pure and Applied Mathematics. 2016; 12(3): 2067-2074.
[21] Birkhoff G. Lattice Theory. American Mathematical Society; 1940.
[22] Gratzer G. Lattice Theory (First Concepts and Distributive Lattice). San Francisco; 1970.
[23] Kuroki N. On fuzzy semigroups. Information Sciences. 1991; 53: 203-236.
[24] Bo Y, Wangming W. Fuzzy ideals on a distributive lattice. Fuzzy Sets and Systems. 1990; 35: 231-240. DOI: https://doi.org/10.1016/0165-0114(90)90196-D
[25] Thomas KV, Nair LS. Intuitionistic fuzzy sublattices and ideals. Fuzzy Information and Engineering. 2011; 3: 321-331. DOI: https://doi.org/10.1007/s12543-011-0086-5
[26] Ajmal N, Thomas KV. Fuzzy lattices. Information Sciences. 1994; 79(3-4): 271-291. DOI: https://doi.org/10.1016/0020-0255(94)90124-4
[27] Ajmal N, Thomas KV. Fuzzy lattices I and II. Journal of Fuzzy Mathematics. 2002; 10(2): 255-296.
[28] Biswas R. Intuitionistic fuzzy subgroups. Mathematical Forum. 1996; 10: 39-44.
[29] Banerjee B, Basnet DK. Intuitionistic fuzzy subring and ideals. Journal of Fuzzy Mathematics. 2003; 11(1): 139-155.
[30] Tripthy BK, Satapathy MK, Choudhury PK. Intuitionistic fuzzy lattices and intuitionistic fuzzy Boolean algebras. International Journal of Engineering and Technology. 2013; 5(3): 2352-2361.
[31] Bharathi P. Picture fuzzy lattices. Advances and Applications in Mathematical Sciences. 2019; 18(10): 1203-1207.
[32] Birkhoff G. Lattice Theory. American Mathematical Society, Providence, RI; 1967.
[33] Dutta P, Ganju S. Some aspects of picture fuzzy set. Transactions of A. Razmadze Mathematical Institute. 2018; 172: 164-175. DOI: https://doi.org/10.1016/j.trmi.2017.10.006