On Γ-Interval Valued Fuzzification of Lagrange’s Theorem of Γ-Interval Valued Fuzzy Subgroups

Abstract

In this paper, we present the idea of interval valued fuzzy subgroup defined over a certain t-conorm ( $\mathrm {\Gamma }$ -IVFSG) and prove that every IVFSG is $\mathrm {\Gamma }$ -IVFSG. We use this ideology to define the concepts of $\Gamma $ -IVF cosets, $\mathrm {\Gamma }$ -IVFNSG and formulate their various important algebraic characteristics. We also propose the study of the notion of level subgroups of $\mathrm {\Gamma }$ -IVFSG and investigate the condition under which a $\mathrm {\Gamma }$ -IVFS is $\Gamma $ -IVFSG. Moreover, we extend the study of this phenomenon to introduce the concept of quotient group of a group $Z$ relative to the $\Gamma $ -IVFNSG and acquire a correspondence between each $\Gamma $ -IVF(N)SG of a group $Z$ and $\Gamma $ -IVF(N)SG of its quotient group. Furthermore, we define the index of $\Gamma $ -IVFSG and establish the $\Gamma $ -interval valued fuzzification of Lagrange’s theorem of any $\Gamma $ -IVFSG of a finite group $Z$ .

Keywords

• Γ-interval valued fuzzy set (Γ-IVFS)
• Γ-interval valued fuzzy subgroup (Γ-IVFSG)
• Γ-interval valued fuzzy coset
• Γ-interval valued fuzzy normal subgroup (Γ-IVFNSG)
• Γ-interval valued fuzzy quotient group