On the fuzzification of Lagrange's theorem in $ (\alpha, \beta) $-Pythagorean fuzzy environment

Supriya Bhunia; Ganesh Ghorai; Qin Xin; Supriya Bhunia; Ganesh Ghorai; Qin Xin

doi:10.3934/math.2021540

AIMS Mathematics

2021, Volume 6, Issue 9: 9290-9308. doi: 10.3934/math.2021540

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Research article

On the fuzzification of Lagrange's theorem in $ (\alpha, \beta) $-Pythagorean fuzzy environment

1.
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721102, India
2.
Faculty of Science and Technology, University of the Faroe Islands, Vestarabryggja 15, FO 100 Torshavn, Faroe Islands

Received: 13 November 2020 Accepted: 07 May 2021 Published: 22 June 2021
MSC : 03E72, 08A72, 20N25

An $ (\alpha, \beta) $-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the $ (\alpha, \beta) $-Pythagorean fuzzy order of elements of groups in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between $ (\alpha, \beta) $-Pythagorean fuzzy order of an element of a group in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for $ (\alpha, \beta) $-Pythagorean fuzzy sets is introduced. The concept of $ (\alpha, \beta) $-Pythagorean fuzzy normalizer and $ (\alpha, \beta) $-Pythagorean fuzzy centralizer of $ (\alpha, \beta) $-Pythagorean fuzzy subgroups are developed. Further, $ (\alpha, \beta) $-Pythagorean fuzzy quotient group of an $ (\alpha, \beta) $-Pythagorean fuzzy subgroup is defined. Finally, an $ (\alpha, \beta) $-Pythagorean fuzzy version of Lagrange's theorem is proved.
Citation: Supriya Bhunia, Ganesh Ghorai, Qin Xin. On the fuzzification of Lagrange's theorem in $ (\alpha, \beta) $-Pythagorean fuzzy environment[J]. AIMS Mathematics, 2021, 6(9): 9290-9308. doi: 10.3934/math.2021540

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Abstract

An $ (\alpha, \beta) $-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the $ (\alpha, \beta) $-Pythagorean fuzzy order of elements of groups in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between $ (\alpha, \beta) $-Pythagorean fuzzy order of an element of a group in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for $ (\alpha, \beta) $-Pythagorean fuzzy sets is introduced. The concept of $ (\alpha, \beta) $-Pythagorean fuzzy normalizer and $ (\alpha, \beta) $-Pythagorean fuzzy centralizer of $ (\alpha, \beta) $-Pythagorean fuzzy subgroups are developed. Further, $ (\alpha, \beta) $-Pythagorean fuzzy quotient group of an $ (\alpha, \beta) $-Pythagorean fuzzy subgroup is defined. Finally, an $ (\alpha, \beta) $-Pythagorean fuzzy version of Lagrange's theorem is proved.

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