1
Department of Mathematics, Federal University of Agriculture, Abeokuta, Nigeria
(agboolaaaa@funaab.edu.ng)
Abstract :
The notion of AntiGroups is formally presented in this paper. A particular class of AntiGroups of type-AG[4] is studied with several examples and basic properties presented. In AntiGroups of type-AG[4], the existence of an inverse is taking to be totally false for all the elements while the closure law, the existence of identity element, the axioms of associativity and commutativity are taking to be either partially true, partially indeterminate or partially false for some elements. It is shown that some algebraic properties of the classical groups do not hold in the class of AntiGroups of type-AG[4]. Specifically, it is shown that intersection of two AntiSubgroups is not necessarily an AntiSubgroup and the union of two AntiSubgroups may be an AntiSubgroup. Also, it is shown that distinct left(right) cosets of AntiSubgroups of AntiGroups of type-AG[4] do not partition the AntiGroups; and that Lagranges’ theorem and fundamental theorem of homomorphisms of the classical groups do not hold in the class of AntiGroups of type-AG[4].
Keywords :
NeutroGroup , AntiGroup , AntiSubgroup , AntiQuotientGroup , AntiGroupHomomorphism
References :
[1] Agboola, A.A.A., On Finite NeutroGroups of Type-NG[1,2,4], International Journal of Neutrosophic Science (IJNS), Vol. 10 (2), pp. 84-95, 2020. (Doi:10.5281/zenodo.4006602)
[2] Agboola, A.A.A., Ibrahim, M.A. and Adeleke, E.O., Elementary Examination of NeutroAlgebras and AntiAlgebras viz-a-viz the Classical Number Systems (IJNS), International Journal of Neutrosophic Science, Vol. 4 (1), pp. 16-19, 2020. DOI:10.5281/zenodo.3752896.
[3] Agboola, A.A.A., Introduction to NeutroGroups, International Journal of Neutrosophic Science (IJNS),Vol. 6 (1), pp. 41-47, 2020. (DOI: 10.5281/zenodo.3840761).
[4] Agboola, A.A.A., Introduction to NeutroRings, International Journal of Neutrosophic Science (IJNS),Vol. 7 (2), pp. 62-73, 2020. (DOI:10.5281/zenodo.3877121).
[5] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, pp. 87-96, 1986.
[6] Gilbert, L. and Gilbert, J., Elements of Modern Algebra, Eighth Edition, Cengage Learning, USA, 2015.
[7] Rezaei, A. and Smarandache, F., On Neutro-BE-algebras and Anti-BE-algebras (revisited), International Journal of Neutrosophic Science (IJNS), Vol. 4 (1), pp. 08-15, 2020. (DOI: 10.5281/zenodo.3751862).
[8] Smarandache, F., NeutroAlgebra is a Generalization of Partial Algebra, International Journal of Neutrosophic Science, vol. 2 (1), pp. 08-17, 2020.
[9] Smarandache, F., Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited), Neutrosophic Sets and Systems (NSS), vol. 31, pp. 1-16, 2020. DOI: 10.5281/zenodo.3638232.
[10] Smarandache, F., Introduction to NeutroAlgebraic Structures, in Advances of Standard and Nonstandard Neutrosophic Theories, Pons Publishing House Brussels, Belgium, Ch. 6, pp. 240-265, 2019.
[11] Smarandache, F., Neutrosophy, Neutrosophic Probability, Set, and Logic, ProQuest Information & Learning, Ann Arbor, Michigan, USA, 105 p., 1998. http://fs.unm.edu/eBook-Neutrosophic6.pdf (edition online).
[12] Zadeh, L.A., Fuzzy Sets, Information and Control, Vol. 8, pp. 338-353, 1965
| Style | # |
|---|---|
| MLA | A.A.A. Agboola. "Introduction to AntiGroups." International Journal of Neutrosophic Science, Vol. 12, No. 2, 2020 ,PP. 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |
| APA | A.A.A. Agboola. (2020). Introduction to AntiGroups. Journal of International Journal of Neutrosophic Science, 12 ( 2 ), 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |
| Chicago | A.A.A. Agboola. "Introduction to AntiGroups." Journal of International Journal of Neutrosophic Science, 12 no. 2 (2020): 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |
| Harvard | A.A.A. Agboola. (2020). Introduction to AntiGroups. Journal of International Journal of Neutrosophic Science, 12 ( 2 ), 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |
| Vancouver | A.A.A. Agboola. Introduction to AntiGroups. Journal of International Journal of Neutrosophic Science, (2020); 12 ( 2 ): 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |
| IEEE | A.A.A. Agboola, Introduction to AntiGroups, Journal of International Journal of Neutrosophic Science, Vol. 12 , No. 2 , (2020) : 71-80 (Doi : https://doi.org/10.54216/IJNS.120201) |