On the fuzzification of Lagrange's theorem in (α,β)-Pythagorean fuzzy environment

Abstract

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.

Keywords

• (α
• β)-pythagorean fuzzy set
• (α，β)-pythagorean fuzzy subgroup
• (α，β)-pythagorean fuzzy order
• (α，β)-pythagorean fuzzy quotient group
• lagrange's theorem