Fuzzy cosets in AG-groups

Aman Ullah; Muhammad Ibrahim; Tareq Saeed; Aman Ullah; Muhammad Ibrahim; Tareq Saeed

doi:10.3934/math.2022185

AIMS Mathematics

2022, Volume 7, Issue 3: 3321-3344. doi: 10.3934/math.2022185

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Fuzzy cosets in AG-groups

1.
Department of Mathematics, University of Malakand, Chakadar, Dir (L), Pakistan
2.
Department of Basic Sciences and Humanities, CECOS University of IT and Emerging Sciences, Peshawar, Pakistan
3.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 05 August 2021 Accepted: 24 November 2021 Published: 30 November 2021
MSC : 08A72, 17A30

In this paper, the notion of fuzzy AG-subgroups is further extended to introduce fuzzy cosets in AG-groups. It is worth mentioning that if $ A $ is any fuzzy AG-subgroup of $ G $, then $ \mu_{A}(xy) = \mu_{A}(yx) $ for all $ x, \, y\in G $, i.e. in AG-groups each fuzzy left coset is a fuzzy right coset and vice versa. Also, fuzzy coset in AG-groups could be empty contrary to fuzzy coset in group theory. However, the order of the nonempty fuzzy coset is the same as the index number $ [G:A] $. Moreover, the notions of fuzzy quotient AG-subgroup, fuzzy AG-subgroup of the quotient (factor) AG-subgroup, fuzzy homomorphism of AG-group and fuzzy Lagrange's theorem of finite AG-group is also introduced.
- AG-group,
- fuzzy AG-subgroup,
- fuzzy normal AG-subgroup,
- fuzzy coset and fuzzy quotient AG-subgroups
Citation: Aman Ullah, Muhammad Ibrahim, Tareq Saeed. Fuzzy cosets in AG-groups[J]. AIMS Mathematics, 2022, 7(3): 3321-3344. doi: 10.3934/math.2022185

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Abstract

In this paper, the notion of fuzzy AG-subgroups is further extended to introduce fuzzy cosets in AG-groups. It is worth mentioning that if $ A $ is any fuzzy AG-subgroup of $ G $, then $ \mu_{A}(xy) = \mu_{A}(yx) $ for all $ x, \, y\in G $, i.e. in AG-groups each fuzzy left coset is a fuzzy right coset and vice versa. Also, fuzzy coset in AG-groups could be empty contrary to fuzzy coset in group theory. However, the order of the nonempty fuzzy coset is the same as the index number $ [G:A] $. Moreover, the notions of fuzzy quotient AG-subgroup, fuzzy AG-subgroup of the quotient (factor) AG-subgroup, fuzzy homomorphism of AG-group and fuzzy Lagrange's theorem of finite AG-group is also introduced.

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