1
School of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India
(vasantha.wb@vit.ac.in)
2
School of Computer Science and Engineering, Vellore Institute of Technology, Vellore, Tamil Nadu, India
(ilanthenral.k@vit.ac.in)
3
Dept. Math and Sciences, University of New Mexico, Gallup, NM, USA
(smarand@unm.edu)
Abstract :
For the first-time authors study the NeutroAlgebraic structures of the substructures of the semigroups, { , ×},
{ , ×} and { , +} where = {1, 2, …, ¥}. The three substructures of the semigroup studied in the context of NeutroAlgebra are subsemigroups, ideals and groups. The substructure group has meaning only if the semigroup under consideration is a Smarandache semigroup. Further in this paper, all semigroups are only commutative. It is proved the NeutroAlgebraic structure of ideals (and subsemigroups) of a semigroup can be AntiAlgebra or NeutroAlgebra in the case of infinite semigroups built on or = È {0}. However, in the case of S = { , ×}; n a composite number, S is always a Smarandache semigroup. The substructures of them are completely analyzed. Further groups of Smarandache semigroups can only be a NeutroAlgebra and never an AntiAlgebra. Open problems are proposed in the final section for researchers interested in this field of study.
Keywords :
NeutroAlgebra , AntiAlgebra , groupring , NeutrosubAlgebra , Partial Algebra , groups , ideals , Smarandache semigroup.
References :
[1] W. B. Vasantha, Smarandache Semigroups, American Research Press, 2002.
[2] F. Smarandache, “A unifying field in logics. neutrosophy: Neutrosophic probability, set and logic,” 1999.
[3] F. Smarandache, Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited). Infinite
Study, 2019.
[4] F. Smarandache, NeutroAlgebra is a Generalization of Partial Algebra. Infinite Study, 2020.
[5] F. Smarandache and A. Rezaei, “On neutro-be-algebras and anti-be-algebras,” International Journal of
Neutrosophic Science, vol. 4, no. 1, p. 8, 2020.
[6] F. Smarandache and M. Hamidi, “Neutro-bck-algebra,” International Journal of Neutrosophic Science, vol. 8,
no. 2, p. 110, 2020.
[7] A. Agboola, M. Ibrahim, and E. Adeleke, “Elementary examination of neutroalgebras and antialgebras viz-a-viz
the classical number systems,” International Journal of Neutrosophic Science (IJNS), vol. 4, pp. 16–19, 2020.
[8] Kandasamy, V., Kandasamy, I., and Smarandache, F., “NeutroAlgebra of Neutrosophic Triplets using .
Neutrosophic Sets and Systems, vol. 38, pp. 34, 2020.
[9] González Caballero, E., Leyva, M., Estupiñán Ricardo, J., & Batista Hernández, N. (2022). NeutroGroups
Generated by Uninorms: A Theoretical Approach. In F. Smarandache, & M. Al-Tahan (Ed.), Theory and
Applications of NeutroAlgebras as Generalizations of Classical Algebras (pp. 155-179). IGI Global.
https://doi.org/10.4018/978-1-6684-3495-6.ch010
[10] Batista Hernández, N., González Caballero, E., Valencia Cruzaty, L. E., Ortega Chávez, W., Padilla Huarac, C.
F., & Chijchiapaza Chamorro, S. L. (2022). Theoretical Study of the NeutroAlgebra Generated by the Combining
Function in Prospector and Some Pedagogical Notes. In F. Smarandache, & M. Al-Tahan (Ed.), Theory and
Applications of NeutroAlgebras as Generalizations of Classical Algebras (pp. 116-140). IGI Global.
https://doi.org/10.4018/978-1-6684-3495-6.ch008
[11] Kandasamy, I., B., V. W., & Smarandache, F. (2022). NeutroAlgebra of Ideals in a Ring. In F. Smarandache, &
M. Al-Tahan (Ed.), Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras (pp. 260-
273). IGI Global. https://doi.org/10.4018/978-1-6684-3495-6.ch015
[12] Kandasamy, V., Kandasamy, I., and Smarandache, F., “NeutroAlgebra of idempotents in Grouprings.
Neutrosophic Sets and Systems, Accepted, 2022.
[13] Silva Jiménez, D., Valenzuela Mayorga, juan A. ., Esther Roja Ubilla, M. ., & Hernández, N. B. . (2021).
NeutroAlgebra for the evaluation of barriers to migrants’ access in Primary Health Care in Chile based on
PROSPECTOR function. Neutrosophic Sets and Systems, 39, 1-9. Retrieved from
http://fs.unm.edu/NSS2/index.php/111/article/view/1216
[14] Valdivieso, W. V., Rodríguez, J. A. V., Villa, M. F. V., Reyes, E. R. R., & Montero, J. S. N. (2021).
Assessment of Barriers to Access Public Services for Immigrants in Ecuador using a NeutroAlgebra-based Model.
Neutrosophic Sets and Systems, 44, 53.
[15] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic,
Hedge Algebras. And Applications. EuropaNova, 2017.
[16] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “A classical group of neutrosophic triplet groups using
,” Symmetry, vol. 10, no. 6, 2018.
[17] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Neutrosophic duplets of and and their
properties,” Symmetry, vol. 10, no. 8, 2018.
[18] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Algebraic structure of neutrosophic duplets in
neutrosophic rings,” Neutrsophic Sets and Systems, vol. 23, pp. 85–95, 2018.
[19] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Neutrosophic triplets in neutrosophic rings,”
Mathematics, vol. 7, no. 6, 2019.
[20] M. Ali, F. Smarandache, and M. Khan, “Study on the development of neutrosophic triplet ring and
neutrosophic triplet field,” Mathematics, vol. 6, no. 4, 2018.
[21] F. Smarandache and M. Ali, “Neutrosophic triplet group,” Neural Comput and Applic, vol. 29, p. 595-601,
2018.
[22] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Neutrosophic quadruple vector spaces and their
properties,” Mathematics, vol. 7, no. 8, 2019.
[23] X. Zhang, X. Wang, F. Smarandache, T. G. Ja, and T. Lian, “Singular neutrosophic extended triplet groups and
generalized groups,” Cognitive Systems Research, vol. 57, pp. 32–40, 2019.
[24] X. Zhou, P. Li, F. Smarandache, and A. M. Khalil, “New results on neutrosophic extended triplet groups
equipped with a partial order,” Symmetry, vol. 11, no. 12, p. 1514, 2019.
[25] X. Zhang, F. Smarandache, and X. Liang, “Neutrosophic duplet semi-group and cancellable neutrosophic
triplet groups,” Symmetry, vol. 9, no. 11, p. 275, 2017.
[26] X. Zhang, X. Wu, X. Mao, F. Smarandache, and C. Park, “On neutrosophic extended triplet groups (loops) and
abel-grassmanna’s groupoids (ag-groupoids),” Journal of Intelligent & Fuzzy Systems, no. Preprint, pp. 1–11, 2019.
[27] X. Zhang, X. Wu, F. Smarandache, and M. Hu, “Left (right)-quasi neutrosophic triplet loops (groups) and
generalized be-algebras,” Symmetry, vol. 10, no. 7, 2018.
[28] X. Zhang, Q. Hu, F. Smarandache, and X. An, “On neutrosophic triplet groups: Basic properties, nt-subgroups,
and some notes,” Symmetry, vol. 10, no. 7, 2018.
[29] Y. Ma, X. Zhang, X. Yang, and X. Zhou, “Generalized neutrosophic extended triplet group,” Symmetry, vol.
11, no. 3, 2019.
[30] A. Rezaei and F. Smarandache, “The neutrosophic triplet of bi-algebras,” Neutrosophic Sets and Systems, vol.
33, no. 1, p. 20, 2020.
[31] W. B. Vasantha, I. Kandasamy, and F. Smarandache, Neutrosophic Triplet Groups and Their Applications to
Mathematical Modelling. EuropaNova: Brussels, Belgium, 2017, 2017.
[32] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Neutrosophic components semigroups and multiset
neutrosophic components semigroups,” Symmetry, vol. 12, no. 5, p. 818, 2020.
[33] W. B. Vasantha, I. Kandasamy, and F. Smarandache, “Neutrosophic quadruple algebraic codes over Z2 and
their properties,” Neutrosophic Sets and Systems, vol. 33, no. 1, p. 12, 2020.
[34] E. Adeleke, A. Agboola, and F. Smarandache, Refined neutrosophic rings II Infinite Study, 2020.
Received: Month Day, Year. Accepted: Month Day, Year
| Style | # |
|---|---|
| MLA | Vasantha Kandasamy , Ilanthenral Kandasamy, Florentin Smarandache. "NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+." International Journal of Neutrosophic Science, Vol. 18, No. 3, 2022 ,PP. 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |
| APA | Vasantha Kandasamy , Ilanthenral Kandasamy, Florentin Smarandache. (2022). NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+. Journal of International Journal of Neutrosophic Science, 18 ( 3 ), 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |
| Chicago | Vasantha Kandasamy , Ilanthenral Kandasamy, Florentin Smarandache. "NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+." Journal of International Journal of Neutrosophic Science, 18 no. 3 (2022): 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |
| Harvard | Vasantha Kandasamy , Ilanthenral Kandasamy, Florentin Smarandache. (2022). NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+. Journal of International Journal of Neutrosophic Science, 18 ( 3 ), 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |
| Vancouver | Vasantha Kandasamy , Ilanthenral Kandasamy, Florentin Smarandache. NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+. Journal of International Journal of Neutrosophic Science, (2022); 18 ( 3 ): 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |
| IEEE | Vasantha Kandasamy, Ilanthenral Kandasamy, Florentin Smarandache, NeutroAlgebra of Substructures of the Semigroups built using Zn and Z+, Journal of International Journal of Neutrosophic Science, Vol. 18 , No. 3 , (2022) : 135-156 (Doi : https://doi.org/10.54216/IJNS.1803012) |