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On Some Properties of Multigroups
Subject area: Science,Engineering and Technology · Area of research: Mathematics (Abstract Algebra)
DOI: https://doi.org/10.64388/IREV9I3-1710873-1146
Abstract
In this paper, we studied some algebraic properties of multigroups. Some of the properties of multisets have also been studied. We discovered that for any multiset G to be a multigroup over the group X, then its count function C_G must satisfies the two conditions: (i) C_G (xy)?[C_G (x)? C_G (y)],? x,y?X; (where ? is the minimum operation). (ii) C_G (x^(-1) )?C_G (x), ? x?X. We also discovered that if X is a group and A,B?MG(X), then A?B?MG(X) but A?B?MG(X). It has also been shown that if A and B are two multigroups over a group X, then A is said to be a submultigroup of B if A?B. Hence, the study shows that the theory of multisets and multigroups can be very useful in many areas such as information retrieval on the web, data mining, decision making, data encryption, coding theory etc.
Keywords
Multisets, Group, Multigroups, Count function.
References
[1] Arfur, D. (1998). An introduction to group theory. http://www.marlboro.edu/~mahoney/groups/dog_school/groups.html.
[2] Awolola, J. A. and Ibrahim, A. M. (2016). Some results on multigroups. Quasigroups and related systems. 24:169-177.
[3] Erno, R. (1974). Group theory. Wikipedia.
[4] Nazmul, S. K., Majumdar, P. and Samanta, S. K. (2013). On multisets and multigroups. Annals of fuzzy mathematics and informatics. 6(3): 643-656.
[5] Marty, F. (1934). Sur une generalization de la notion de groupe. Huitieme congres des mathematiciens scandinaves, Stockholm. 201:45-49.
[6] Melvin, D. and Oystein, O. (1938). Theory of multigroups. American journal of mathematics. 60(3): 705-733.
[7] Shinoj, T. K. and Sunil, J. J. (2015). Intuitionistic fuzzy multigroups. Annals of pure and applied mathematics. 9: 131-143.
[8] Wall, H. S. (1937). Hypergroups. American journal of mathematics. 59:77-98.
How to cite this paper
@article{1710873,
author = {Ewaoche Nicholas, Samuel Henry Owoicho, Okpanachi Sunday, Uko Benjamin Joseph, Okpanachi George Echiye},
title = {On Some Properties of Multigroups},
journal = {Iconic Research And Engineering Journals},
year = {2025},
volume = {9},
number = {3},
pages = {1485-1490},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1710873.pdf},
abstract = {In this paper, we studied some algebraic properties of multigroups. Some of the properties of multisets have also been studied. We discovered that for any multiset G to be a multigroup over the group X, then its count function C_G must satisfies the two conditions: (i) C_G (xy)?[C_G (x)? C_G (y)],? x,y?X; (where ? is the minimum operation). (ii) C_G (x^(-1) )?C_G (x), ? x?X. We also discovered that if X is a group and A,B?MG(X), then A?B?MG(X) but A?B?MG(X). It has also been shown that if A and B are two multigroups over a group X, then A is said to be a submultigroup of B if A?B. Hence, the study shows that the theory of multisets and multigroups can be very useful in many areas such as information retrieval on the web, data mining, decision making, data encryption, coding theory etc.},
keywords = {Multisets, Group, Multigroups, Count function.},
month = {September},
doi = {https://doi.org/10.64388/IREV9I3-1710873-1146}
}