On Normal and Soft Normal Groups under Multisets and Soft Multiset Context

Idris Muhammadu Adamu, Y. Tella, A. J. Alkali


In this paper, the concepts of normal submultigroups and soft normal multigroups with some of their related algebraic structures were developed. We established that intersection, union of any two normal submultigroups is also normal submultigroup of a given multigroup. The inverse of any normal submultigroup of a multigroup is a normal submultigroup and for any normal submultigroup, the root (support) set is a normal subgroup of the underlying group.  We then showed that under the isomorphism function between any two groups, the image of a normal submultigroup under the isomorphism is a normal submultigroup and the inverse image of a normal submultigroup under the isomorphism is a normal submultigroup. Finally, we defined operations on soft normal multigroups such as intersection, union, AND, OR operations and discovered that such operations are closed under soft normal multigroups.

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D.E. Knuth, The Art of computer programming, vol. 2: Semi numerical algorithms. Adison Wesley, 1981.

N. G. De Bruijin, "Denumerations of Rooted trees and Multisets," Descrete Appl. Math, vol. 6, pp. 25-33, 1983.

D. Singh, A. M. Ibrahim, T. Yohana and J. N. Sing, "An overview of the applications of multisets," Novi Sad J. Math., vol. 37, no. 2, pp. 73-92, 2007.

R. R. Yager, "On the theory of bags," Int. J. Gen. Sys., vol. 13, pp. 23-37, 1986.

W. D. Blizard, "Multiset theory," Notre Dame J. Form. Log, vol. 30, pp. 36-66, 1989.

K. Girish and S. J. John, "Relations and Functions in Multiset Context," Inform. Sci., vol. 176(6), pp. 758-768, 2009.

K. P. Girish and S. J. John, "General Relations between Partially Ordered Multisets and their Chains and Antichains," Maths Communication, vol. 14(2), pp. 193-206, 2009.

D. Molodtsov, "Soft Set Theory-First Result," Comput. Math. Appl, vol. 37, pp. 19-31, 1999.

P. K. Maji, R. Biswas and A. Roy, "An application of soft sets in decision making problem," Comput. Math. Appl., vol. 44, pp. 1077-1083, 2002.

P. K. Maji, R. Biswas and A. Roy, "Soft Set Theory," Comput. Math. Appl, vol. 45, pp. 555-562, 2003.

M. D. Mustafa and T. Herawan, "On Multi-Soft Sets Construction in Information System," in ICIC 2009 LNAI, Springer, Heidelberg 5755, pp.101-110, 2009.

S. Alkhadeh, A. R. Salleh and N. Hassan, "Soft Multiset Theory," Appl. Math. Sci., vol. 5, pp. 3561-3573, 2011.

P. Mujumdar and S. K. Samanta, "On Soft Multiset," Journal of Mathematics and Computational Sciences, vol. 2, no. 6, pp. 1700-1711, 2012.

A. Klausner and N. Goodman, "On Multirelations-Semantics and Languagees," in Procedings of the 11th Conferences on Very Large Data Base VLDB 85, 1985.

I. S. Mumick, H. Pirahesh and Ramakrishnan, "The magic of duplicates and aggregate," in Proceedings of the 6th Conference on Very Large Data Base VLDB 90, 1990.

G. Paun and M. J. Perez-Jimenez, "Membrane Computing: brief introduction, resent results and applications," Bio Systems, vol. 85, pp. 271-285, 2006.

T. Yohannah and S. Daniel, “Symmetric Groups under Multisets context," IORS Journal of Mathematics 7(5), pp. 47-52, 2013.

Y. Tella and S. Daniel, “Study of Group Theory in the Context of Multiset Theory," International Journal of Science and Technology 2(8), pp. 609-615, 2013.

S. K. Nazmul, P. Majumdar and S. Samanta, “Multisets and multigroups," Ann. Fuzzy Math. Inform, vol. 6, no. 3, pp. 643-656, 2013.

S. K. Nazmul and S. K. Samanta, "On soft multigroups," Annals of Fuzzy Mathematics and Informatics, Volume 10 No. 2, pp. 271-285, 2015.

A. Kharal and B. Ahmad, "Mappings on Soft Classes," New Math. Nat. Comput, vol.7, pp. 471-481, 2011.

D. Tokat and I. Osmanogulu, "On Connectedness on Soft Multi Topological Spaces," Journal of New Result in Sciences, vol. 2, pp. 8-18, 2013.

B. Li, W. Peizhang and L. Xihui, “Fuzzy bags with set-valued statistics," Comput. Math. Appl., vol. 15, pp. 3-39, 1988.

T. K. Shinoj and S. J. John, “Intuitionistic Fuzzy Multigroups," Annal of Pure and Applied Mathematics, Vol. 9, No. 1, pp.

-143, 2015.

T. K. Shinoj, A. Baby and S. J. John, “on some algebraic structures of fuzzy multisets," Annl. Fuzzy Math. Inform. Vol. 9. No.1,

pp. 77-90, 2015.

S. K. Nazmul and S. K. Samanta, " Fuzzy soft group," J. Fuzzy Math., vol. 19, no.1, pp. 101-114, 2011.

H. Aktas and N. Cagman, “Soft sets and soft groups,” Inform. Sci., vol. 177, pp. 2726-2735, 2007.

YU Feng, “Intuitionistic L-fuzzy groups,” School of Power Engineering, Nanjing University of Sciences and Technology,

Nanjing 210094, P. R. China.

A. Rosenfeld, “Fuzzy Groups," J. Math. Appl., vol. 35, pp. 512-517, 1971.

L. A. Zadeh, “Fuzzy sets,” Inform. Control, vol. 8, pp. 338-353, 1965.


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