𝒩-Structures Applied to Commutative Ideals of BCI-Algebras
Abstract
:1. Introduction
- In 2018, Jun et al. inroduced the notion of Neutrosophic positive “implicative -I” in -algebras [27].
- In 2019, Muhiuddin et al. studied “implicative -I” of BCK-algebras based on “Neutrosophic -STRUC” [28].
- Most recently, in 2021, Muhiuddin initiated the concept of “p-BCI-ideals” based on “Neutrosophic -STRUC” [29].
2. Preliminaries
- (a1)
- (a2)
- (a3)
- (a4)
- (b1)
- (b2)
- (b3)
- (b4)
- (a5)
- for all
- (a6)
- (a7)
- (a8)
- (a9)
- (a10)
- for all
- (c1)
- (c2)
- (c3)
- (a)
- is an “-SUB” of
- (b)
- ∀.
- (a)
- is an “-I” of
- (b)
- ∀.
3. BCI-Commutative -Ideals
- (c4)
- (1)
- (2)
- (3)
- (4)
- (5)
- (6)
- (7)
- (8)
- the BCK-part of Ω is
- (d1)
- (1)
- is a “COMMU -I” of
- (2)
- (d2)
- (a)
- is a “COMMU -I” of
- (b)
- satisfies the following inequality:
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Imai, Y.; Iséki, K. On axiom systems of propositional calculi. Proc. Jpn. Acad. 1966, 42, 19–21. [Google Scholar] [CrossRef]
- Iséki, K. An algebra related with a propositional calculus. Proc. Jpn. Acad. 1966, 42, 26–29. [Google Scholar] [CrossRef]
- Borzooei, R.A.; Smarandache, F.; Jun, Y.B. Polarity of generalized neutrosophic subalgebras in BCK/BCI-algebras. Neutrosophic Sets Syst. 2020, 32, 123–145. [Google Scholar]
- Huang, Y.S. BCI-Algebra; Science Press: Beijing, China, 2006. [Google Scholar]
- Meng, J.; Jun, Y.B. BCK-Algebras; Kyung Moon Sa: Seoul, Korea, 1994. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef]
- Jun, Y.B.; Lee, K.J.; Song, S.Z. -ideals of BCK/BCI-algebras. J. Chungcheong Math. Soc. 2009, 22, 417–437. [Google Scholar]
- Feng, F.; Fujita, H.; Ali, M.I.; Yager, R.R.; Liu, X. Another View on Generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods. IEEE Trans. Fuzzy Syst. 2019, 27, 474–488. [Google Scholar] [CrossRef]
- Ali, M.I.; Feng, F.; Mahmood, T.; Mahmood, I.; Faizan, H. A graphical method for ranking Atanassov’s intuitionistic fuzzy values using the uncertainty index and entropy. Int. J. Intell. Syst. 2019, 34, 2692–2712. [Google Scholar] [CrossRef]
- Aliev, R.A.; Pedrycz, W.; Huseynov, O.H.; Eyupoglu, S.Z. Approximate Reasoning on a Basis of Z-Number-Valued If–Then Rules. IEEE Trans. Fuzzy Syst. 2017, 25, 1589–1600. [Google Scholar] [CrossRef]
- Tang, Y.; Pedrycz, W. Oscillation-Bound Estimation of Perturbations Under Bandler–Kohout Subproduct. IEEE Trans. Cybern. 2022, 52, 6269–6282. [Google Scholar] [CrossRef]
- Jun, Y.B.; Kang, M.S.; Park, C.H. N-subalgebras in BCK/BCI-algebras based on point N-structures. Int. J. Math. Math. Sci. 2010, 2010, 303412. [Google Scholar] [CrossRef]
- Jun, Y.B.; Öztürk, M.A.; Roh, E.H. -structures applied to closed ideals in BCH-algebras. Int. J. Math. Math. Sci. 2010, 2010, 943565. [Google Scholar] [CrossRef]
- Jun, Y.B.; Kavikumar, J.; So, K.S. N-ideals of subtraction algebra. Commun. Korean Math. Soc. 2010, 25, 173–184. [Google Scholar] [CrossRef] [Green Version]
- Jun, Y.B.; Lee, K.J. The essence of subtraction algebras based on N-structures. Commun. Korean Math. Soc. 2012, 27, 15–22. [Google Scholar] [CrossRef]
- Jun, Y.B.; Lee, K.J.; Kang, M.S. Ideal Theory in BCK/BCI-Algebras Based on Soft Sets and N-Structures. Discrete Dyn. Nat. Soc. 2012, 2012, 910450. [Google Scholar] [CrossRef]
- Lee, K.J.; Jun, Y.B.; Zhang, X. N-subalgebras of type (∈,∈∨q) based on point N-structures in BCK/BCI-algebras. Commun. Korean Math. Soc. 2012, 27, 431–439. [Google Scholar] [CrossRef]
- Jun, Y.B.; Kang, M.S. Ideal thoery of BE-algebras based on N-structures. Hacet. J. Math. Stat. 2012, 41, 435–447. [Google Scholar]
- Ejegwa, P.A.; Otuwe, J.A. Frattini fuzzy subgroups of fuzzy groups. Ann. Commun. Math. 2019, 2, 24–31. [Google Scholar]
- Muhiuddin, G. Neutrosophic Subsemigroups. Ann. Commun. Math. 2018, 1, 1–10. [Google Scholar]
- Muhiuddin, G.; Porselvi, K.; Elavarasan, B.; Al-Kadi, D. Neutrosophic N-Structures in Ordered Semigroups. Comput. Model. Eng. Sci. 2022, 131, 979–999. [Google Scholar] [CrossRef]
- Senapati, T.; Shum, K.P. Cubic subalgebras of BCH-algebras. Ann. Commun. Math. 2018, 1, 65–73. [Google Scholar]
- Jun, Y.B.; Ahn, S.S. Applications of Coupled N-structures in BCC-Algebras. J. Comput. Anal. Appl. 2014, 16, 740–749. [Google Scholar]
- Jun, Y.B.; Alshehri, N.O.; Lee, K.J. Soft set theory and N-structures applied to BCH-algebras. J. Comput. Anal. Appl. 2014, 16, 869–886. [Google Scholar]
- Khan, M.; Anis, S.; Smarandache, F.; Jun, Y.B. Neutrosophic -structures and their applications in semigroups. Ann. Fuzzy Math. Inform. 2017, 14, 583–598. [Google Scholar] [CrossRef]
- Song, S.Z.; Smarandache, F.; Jun, Y.B. Neutrosophic commutative -ideals in BCK-algebras. Information 2017, 8, 130. [Google Scholar] [CrossRef]
- Jun, Y.B.; Smarandache, F.; Song, S.Z.; Khan, M. Neutrosophic positive implicative -ideals in BCK-algebras. Axioms 2018, 7, 3. [Google Scholar] [CrossRef]
- Muhiuddin, G.; Kim, S.J.; Jun, Y.B. Implicative N-ideals of BCK-algebras based on neutrosophic N-structures. Discret. Math. Algorithms Appl. 2019, 11, 1950011. [Google Scholar] [CrossRef]
- Muhiuddin, G. p-ideals of BCI-algebras based on neutrosophic N-structures. J. Intell. Fuzzy Syst. 2021, 40, 1097–1105. [Google Scholar] [CrossRef]
- Meng, J.; Xin, X.L. Commutative BCI-algebras. Math. Japon. 1992, 37, 569–572. [Google Scholar]
- Meng, J. An ideal characterization of commutative BCI-algebra. Pusan Kyongnam Math. J. 1993, 9, 1–6. [Google Scholar]
- Jun, Y.B.; Kondo, M. On transfer principle of fuzzy BCK/BCI-algebras. Sci. Math. Jpn. 2004, 59, 35–40. [Google Scholar]
- Kondo, M.; Dudek, W.A. On the transfer principle in fuzzy theory. Mathware Soft Comput. 2005, 12, 41–55. [Google Scholar]
- Bhatti, S.A.; Chaudhry, M.A.; Ahmad, B. On classification of BCI-algebras. Math. Japon. 1989, 34, 865–876. [Google Scholar]
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Muhiuddin, G.; Elnair, M.E.; Al-Kadi, D. 𝒩-Structures Applied to Commutative Ideals of BCI-Algebras. Symmetry 2022, 14, 2015. https://doi.org/10.3390/sym14102015
Muhiuddin G, Elnair ME, Al-Kadi D. 𝒩-Structures Applied to Commutative Ideals of BCI-Algebras. Symmetry. 2022; 14(10):2015. https://doi.org/10.3390/sym14102015
Chicago/Turabian StyleMuhiuddin, Ghulam, Mohamed E. Elnair, and Deena Al-Kadi. 2022. "𝒩-Structures Applied to Commutative Ideals of BCI-Algebras" Symmetry 14, no. 10: 2015. https://doi.org/10.3390/sym14102015
APA StyleMuhiuddin, G., Elnair, M. E., & Al-Kadi, D. (2022). 𝒩-Structures Applied to Commutative Ideals of BCI-Algebras. Symmetry, 14(10), 2015. https://doi.org/10.3390/sym14102015