ON NEW INTUITIONISTIC FUZZY(\ep,η)-NEGATION AND (\ep,η)-IMPLICATION

Abstract

The concept of an Intuitionistic Fuzzy Set (IFS) was introduced in 1983 in \cite{ITKR} and again there in it, the first definition of operation intuitionistic fuzzy negation over IFS was introduced. Later, in \cite{IFPC}, two forms of the operation intuitionistic fuzzy implication were introduced. As of the 2000s, more than 200 different implications and more than 50 different negations arise and in [5 -- 21, 23, 24], their basic properties were discussed. In \cite{e,ee}, two negations, namely $\ep$-negation and $(\ep,\eta)$-negation, were introduced and over their bases $\ep$- and $(\ep,\eta)$-implications were defined, respectively. It was shown that the $(\ep,\eta)$-negation and $(\ep,\eta)$-implication are extensions of the $\ep$-negation and $\ep$-implication, respectively.

Keywords

• Implication
• Intuitionistic fuzzy set
• Negation

References

1. \bibitem{ITKR} Atanassov, K. Intuitionistic fuzzy sets, VII ITKR's Session, Sofia, June 1983 (Deposed in Central Sci. - Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulg.). Reprinted: \textit{Int. J. Bioautomation}, 2016, 20(S1), S1-S6. \bibitem{IFPC} Atanassov K., Two variants of intuitionistic fuzzy propositional calculus.\textit{Preprint IM-MFAIS-5-88}, Sofia, 1988. Reprinted: \textit{Int. J. Bioautomation}, 2016, 20(S1), S17-S26.
2. \bibitem{e} Atanassov, K. On intuitionistic fuzzy implication $\ri^{\ep}$ and intuitionistic fuzzy negation $\neg^{\ep}$. \textit{Issues in Intuitionistic Fuzzy Sets and Generalized Nets}, Vol. 6, 2008, 6-19.
3. \bibitem{ee} Atanassov, K. Intuitionistic fuzzy implication $\riee$ and intuitionistic fuzzy negation $\nee$. \textit{Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics}, Vol. 1, 2008, 1-10. \bibitem{Spr2} Atanassov, K. (2012) \textit{On Intuitionistic Fuzzy Sets Theory}, Springer, Berlin. \bibitem{Spr4} Atanassov, K. (2017) \textit{Intuitionistic Fuzzy Logics}, Springer, Cham.
4. \bibitem{L1} Atanassova, L. (2008). On an intuitionistic fuzzy implication from Kleene-Dienes type. \textit{Proceedings of the Jangjeon Mathematical Society}, 11 (1), 69-74.
5. \bibitem{L4} Atanassova, L. (2008). Modifications of an intuitionistic fuzzy implication from Kleene--Dienes type. \textit{Advanced Studies in Contemporary Mathematics}, 16 (2), 155-160.
6. \bibitem{L5} Atanassova, L. (2008). New modifications of an intuitionistic fuzzy implication from Kleene--Dienes type. Part 2. \textit{Annual of Section ``Informatics"}, 1, 59-64.
7. \bibitem{L6} Atanassova, L. (2009). New modifications of an intuitionistic fuzzy implication from Kleene--Dienes type. Part 3. \textit{Advanced Studies in Contemporary Mathematics}, 18 (1), 33-40.
8. \bibitem{L8} Atanassova, L. (2009). A new intuitionistic fuzzy implication, \textit{Cybernetics and Information Technologies}, 9 (2), 21-25. \bibitem{L9} Atanassova, L. (2009). On some properties of intuitionistic fuzzy negation $\neg_@$, \textit{Notes on Intuitionistic Fuzzy Sets}, 15 (1), 32-35.

9. \bibitem{L15} Atanassova, L. (2012). On two modifications of the intuitionistic fuzzy implication $\ri_@$, \textit{Notes on Intuitionistic Fuzzy Sets}, 18 (2), 26-30
10. \bibitem{L18} Atanassova, L. (2013). On the modal form of the intuitionistic fuzzy implications $\ri'_@$ and $\ri_@''$, \textit{Issues in Intuitionistic Fuzzy Sets and Generalized Nets}, 10, 5-11.
11. \bibitem{L22} Atanassova, L. (2015). Remark on Dworniczak's intuitionistic fuzzy implications. Part 1, \textit{Notes on Intuitionistic Fuzzy Sets}, 21 (3), 18-23.
12. \bibitem{L23} Atanassova, L. (2015/2016) Remark on Dworniczak's intuitionistic fuzzy implications. Part 2, \textit{Issues in Intuitionistic Fuzzy Sets and Generalized Nets}, Vol. 12, 61-67.
13. \bibitem{L24} Atanassova, L. (2016). Remark on Dworniczak's intuitionistic fuzzy implications. Part 3, \textit{Notes on Intuitionistic Fuzzy Sets}, 22 (1), 1-6.
14. \bibitem{L11} Atanassova, L. (2017) Intuitionistic fuzzy implication $\ri_{189}$. \textit{Notes on Intuitionistic Fuzzy Sets}, 23, 1, 14-20.
15. \bibitem{P1} Dworniczak, P. (2010). Some remarks about the L. Atanassova's paper ``A new intuitionistic fuzzy implication", \textit{Cybernetics and Information Technologies}, 10 (3), 3-9.
16. \bibitem{P2} Dworniczak, P. (2010). On one class of intuitionistic fuzzy implications,\textit{Cybernetics and Information Technologies}, 10 (4), 13-21.
17. \bibitem{P3} Dworniczak, P. (2011). On some two-parametric intuitionistic fuzzy implication, \textit{Notes on Intuitionistic Fuzzy Sets}, 17 (2), 8-16.
18. \bibitem{K&Y} Klir, G., B. Yuan, \textit{Fuzzy Sets and Fuzzy Logic}. Prentice Hall, New Jersey, 1995.
19. \bibitem{PV} Vassilev, P., K. Atanassov. \textit{Extensions and Modifications of Intuitionistic Fuzzy Sets}. ``Prof. Marin Drinov" Academic Publishing House, Sofia, 2019.
20. \bibitem{PSJ} Vassilev, P., S. Ribagin, J. Kacprzyk (2018). A remark on intuitionistic fuzzy implications. \textit{Notes on Intuitionistic Fuzzy Sets}, 24 (2), 1-7.