On Singh’s dressed epsilon perspective of multigroup
DOI: https://doi.org/10.33003/jobasr
Chinedu M. Peter
Abstract
This paper investigates some aspects of multigroups from Singh's perspective. It introduces a structured approach to analyzing the aspects of multigroups presented in this paper using dressed epsilon notation. We begin by defining the hierarchical decomposition of multisets, establishing that each r-level reference set in the hierarchical decomposition of a multiset over a group is itself a subgroup. We then present a fundamental characterization of multigroups by proving that a multiset A is a multigroup over a set X if and only if xy^(-1) ∈^p A,∀x∈^m A⟹p≥(m∧n). Additionally, we define the sets A^* and A_* and prove that both are subgroups of X using Singh’s dressed epsilon notation. Our work further investigates the algebraic properties of multigroups and establishes criteria for commutativity. We also demonstrate that while the intersection of two multigroups is always a multigroup, their union does not necessarily inherit this structure. The concept of submultigroup is introduced to formalize the relationship between two multigroups. Finally, we establish the equivalence between certain multigroup properties, such as the symmetry of multisets based on product of elements and conjugate conditions.
References
Awolola, Johnson Aderemi (2019) On multiset relations and factor multigroups, South East Asian J. of Mathematics and Mathematical Sciences, 15(3), 1-10.
Awolola, Johnson Aderemi and Ejegwa (2017) On some algebraic properties of order of an element of a multigroup, Quasigroups and Related Systems, Vol. 25, 21-26.
Awolola, Johnson Aderemi and Ibrahim, Adeku Musa (2016). Some results on multigroups, Quasigroups and Related Systems Vol. 24, 169-177.
Laible, J. M., Baumslag, B., & Chandler, B. (1969). Theory and Problems of Group Theory. The American Mathematical Monthly, 76(6), 712. https://doi.org/10.2307/2316717
Singh, D. (2006). Multiset Theory: A New Paradigm of Science: an Inaugural Lecture. University Organized Lectures Committee, Ahmadu Bello University.
Singh, D., Ibrahim, A. M., Yohanna, T., & Singh, J. N. (2008). A systematization of fundamentals of multisets. Lecturas Matemáticas, 29, 33–48.
Singh, D., & Isah, A. I. (2016). Mathematics of Multisets: a unified Approach. Afrika Matematika, 27(7), 1139-1146. Chicago
Ejegwa, P. A., & Ibrahim, A. M. (2020). Some properties of multigroups. Palestine Journal of Mathematics, 9(1), 31-47.
Ibrahim, A. M., Awolola, J. A., and Alkali, A. J. (2016). An extension of the concept of n-level sets to multisets. Annals of Fuzzy Mathematics and Informatics, 11(6), 855-862.
Nazmul, S. K. Majumdar P. and Samanta S. K. (2013). On Multisets and Multigroups, Ann. Fuzzy Math. Informa. Vol. 6, No. 3, 643–656.
Peter, C., Balogun, F. and Adeyemi, O. A. (2024). an exploration of antimultigroup extensions. fudma journal of sciences, 8(5), 269-273. https://doi.org/10.33003/fjs-2024-0805-2719
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