Volume 19 , Issue 1 , PP: 363-374, 2022 | Cite this article as | XML | Html | PDF | Full Length Article
Aiyared Iampan 1 * , P. Jayaraman 2 , S. D. Sudha 3 , Said Broumi 4 , N. Rajesh 5
Doi: https://doi.org/10.54216/IJNS.190133
Interval-valued neutrosophic sets (IVNSs) are a notion that was initially developed by Wang et al.19 The idea
of IVNSs to deductive systems (DSs) in Hilbert algebras is presented in this study. It is shown how intervalvalued
neutrosophic deductive systems (IVNDSs) relate to their level cuts. In addition, certain related features
are examined as well as the homomorphic inverse image of IVNDSs in Hilbert algebras.
Zadeh20 first developed the idea of fuzzy sets (FSs). Numerous academics have studied FS theory because
it has numerous practical applications. Numerous investigations were undertaken on the generalizations of
FSs after the idea of FSs was introduced. In,1, 3, 6 it is explained how FSs may be integrated with various
uncertainty-reduction strategies like soft sets and rough sets. The idea of intuitionistic fuzzy sets (IFSs),
as out by Atanassov,2 is one of the more beneficial extensions of FSs. Medical diagnostics, optimization
problems, and multi-criteria decision-making are just a few of the areas in which IFSs are applied.10&ndash , 12 In
1999, Smarandache16 presented the idea of neutrosophic sets, which is a broader concept that encompasses the
ideas of classic sets, FSs, IFSs, and interval-valued (I)FSs (see16, 17).
[1] B. Ahmad, A. Kharal, On fuzzy soft sets, Adv. Fuzzy Syst., vol. 2009, Article ID 586507, 6 pages, 2009.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst., vol. 20, no. 1, pp. 87–96, 1986.
[3] M. Atef, M. I. Ali, T. Al-shami, Fuzzy soft covering based multi-granulation fuzzy rough sets and their
applications, Comput. Appl. Math., vol. 40, no. 4, pp. 115, 2021.
[4] D. Busneag, A note on deductive systems of a Hilbert algebra, Kobe J. Math., vol. 2, pp. 29–35, 1985.
[5] D. Busneag, Hilbert algebras of fractions and maximal Hilbert algebras of quotients, Kobe J. Math., vol.
5, pp. 161–172, 1988.
[6] N. Caˇgman, S. Enginoˇglu, F. Citak, Fuzzy soft set theory and its application, Iran. J. Fuzzy Syst., vol. 8,
no. 3, pp. 137–147, 2011.
[7] A. Diego, Sur les alg´ebres de Hilbert, Collection de Logique Math. Ser. A (Ed. Hermann, Paris), vol. 21,
pp. 1–52, 1966.
[8] W. A. Dudek, On fuzzification in Hilbert algebras, Contrib. Gen. Algebra, vol. 11, pp. 77–83, 1999.
[9] W. A. Dudek, On ideals in Hilbert algebras, Acta Universitatis Palackianae Olomuciensis Fac. rer. nat.
ser. Math., vol. 38, pp. 31–34, 1999.
[10] H. Garg, K. Kumar, An advanced study on the similarity measures of intuitionistic fuzzy sets based on
the set pair analysis theory and their application in decision making, Soft Comput., vol. 22, no. 15, pp.
4959–4970, 2018.
[11] H. Garg, K. Kumar, Distance measures for connection number sets based on set pair analysis and its
applications to decision-making process, Appl. Intell., vol. 48, no. 10, pp. 3346–3359, 2018.
[12] H. Garg, S. Singh, A novel triangular interval type-2 intuitionistic fuzzy set and their aggregation operators,
Iran. J. Fuzzy Syst., vol. 15, no. 5, pp. 69–93, 2018.
[13] Y. B. Jun, Deductive systems of Hilbert algebras, Math. Japon., vol. 43, pp. 51–54, 1996.
[14] Y. B. Jun, R. Bandaru, Deductive systems of GE-algebras, Algebr. Struct. Appl., vol. 9, no. 1, pp. 53–67,
2022.
[15] Y. B. Jun, F. Smarandache, C. S. Kim, Neutrosophic cubic sets, New Math. Nat. Comput., vol. 13, no. 1,
pp. 41–54, 2017.
[16] F. Smarandache, A unifying field in logics: Neutrosophic logic, neutrosophy, neutrosophic set, neutrosophic
probability, American Research Press, 1999.
[17] F. Smarandache, Neutrosophic set, a generalization of intuitionistic fuzzy sets, Int. J. Pure Appl. Math.,
vol. 24, no. 5, pp. 287–297, 2005.
[18] K. Taboon, P. Butsri, A. Iampan, A cubic set theory approach to UP-algebras, J. Interdiscip. Math., vol.
23, no. 8, pp. 1449–1486, 2020.
[19] H. Wang, F. Smarandache, Y. Q. Zhang, R. Sunderraman, Interval neutrosophic sets and logic: Theory
and applications in computing, Hexis, Phoenix, Ariz, USA, 2005.
[20] L. A. Zadeh, Fuzzy sets, Inf. Control, vol. 8, no. 3, pp. 338–353, 1965.
[21] J. Zhan, Z. Tan, Intuitionistic fuzzy deductive systems in Hibert algebra, Southeast Asian Bull. Math.,
vol. 29, no. 4, pp. 813–826, 2005.
