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

• Published in: AIMS mathematics Vol. 6; no. 9; pp. 9290 - 9308
• Main Authors: Bhunia, Supriya, Ghorai, Ganesh, Xin, Qin
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2021
• Subjects: (α，β)-pythagorean fuzzy order; (α，β)-pythagorean fuzzy quotient group; (α，β)-pythagorean fuzzy subgroup; lagrange's theorem; β)-pythagorean fuzzy set
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2021540

An (α,β)-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 (α,β)-Pythagorean fuzzy order of elements of groups in (α,β)-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between (α,β)-Pythagorean fuzzy order of an element of a group in (α,β)-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for (α,β)-Pythagorean fuzzy sets is introduced. The concept of (α,β)-Pythagorean fuzzy normalizer and (α,β)-Pythagorean fuzzy centralizer of (α,β)-Pythagorean fuzzy subgroups are developed. Further, (α,β)-Pythagorean fuzzy quotient group of an (α,β)-Pythagorean fuzzy subgroup is defined. Finally, an (α,β)-Pythagorean fuzzy version of Lagrange's theorem is proved.