Picture Fuzzy Ideals in Semigroups under the Type-3 Order: Bi-Ideals, Quasi-Ideals, and Interior Ideals

Abstract

This paper studies picture fuzzy subsets of semigroups under the order (D^*,≤_3 ). We first clarify the admissible domain D^*={(φ,ψ,ω)∈[0,1]^3:0≤φ+ψ+ω≤1} and then redefine basic operations (intersection, product, Cartesian product, image, inverse image, and characteristic function) so that all results remain inside the picture fuzzy framework. Based on these operations, we define picture fuzzy subsemigroups, bi-ideals, quasi-ideals, and interior ideals, and we prove their fundamental closure and characterization properties. We also provide counterexamples showing why some earlier formulations can fail to produce valid picture fuzzy sets. Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.

Keywords

• Picture fuzzy ideals
• Semigroups
• Picture fuzzy sets

Ethical Statement

Ethics committee approval was not required for this study because of there was no study on animals or humans.

References

1. Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3
2. Clifford, A. H., & Preston, G. B. (1961). The algebraic theory of semigroups. American Mathematical Society.
3. Cuong, B. C. (2013). Picture fuzzy sets: First results part 1 (Preprint 03/2013). Seminar on Neuro-Fuzzy Systems and Applications.
4. Cuong, B. C., Kreinovich, V., & Ngan, R. T. (2016). A classification of representable t-norm operators for picture fuzzy sets. Proceedings of the International Conference on Knowledge and Systems Engineering, 21–26. https://doi.org/10.1109/KSE.2016.7758023
5. Dogra, S., & Pal, M. (2022). Ideals of a multiplicative semigroup in picture fuzzy environment. Palestine Journal of Mathematics, 11(4), 215–224.
6. Dogra, S., & Pal, M. (2023). Picture fuzzy subgroup. Kragujevac Journal of Mathematics, 47(6), 911–933. https://doi.org/10.46793/KgJMat2306.911D
7. Grätzer, G. (2002). General lattice theory. Springer.
8. Klement, E. P., Mesiar, R., & Stupňanová, A. (2018). Picture fuzzy sets and 3-fuzzy sets. IEEE International Conference on Fuzzy Systems, 1–6. https://doi.org/10.1109/FUZZ-IEEE.2018.8491585

9. Okumus, E. F., Yamak, S., & Davvaz, B. (baskıda). Redefined picture fuzzy ideals on semigroups. Journal of Multiple-Valued Logic and Soft Computing.
10. Parimala, M., Muchtadi-Alamsyah, I., & Al-Tahan, M. (2021). On picture fuzzy ideals of near-rings. International Journal of Fuzzy Logic and Intelligent Systems, 21(3), 259–268. https://doi.org/10.5391/IJFIS.2021.21.3.259
11. Rosenfeld, A. (1971). Fuzzy groups. Journal of Mathematical Analysis and Applications, 35(3), 512–517. https://doi.org/10.1016/0022-247X(71)90199-5
12. Yiarayong, P. (2020). Semigroup characterized by picture fuzzy set. International Journal of Innovative Computing, Information and Control, 16(6), 2121–2130. https://doi.org/10.24507/ijicic.16.06.2121
13. Yiarayong, P. (2022a). A new approach of fuzzy bi-ideal theory applied on semigroups. Soft Computing, 26, 4227–4236. https://doi.org/10.1007/s00500-022-06852-1
14. Yiarayong, P. (2022b). Characterisations of semigroups by the properties of their picture fuzzy bi-ideals. Journal of Control and Decision, 9(1), 111–116. https://doi.org/10.1080/23307706.2021.1919229
15. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X