Some innovative types of fuzzy ideals in AG-groupoids

Abstract

AG-groupoids (non-associative structure) are basic structures in Flocks theory. This theory mainly focuses on distance optimization, motion replication, and leadership maintenance with a wide range of applications in physics and biology. In this paper, we define some new types of fuzzy ideals of AG-groupoids called (α, β)-fuzzy bi-ideals, (α, β)-fuzzy interior ideals, (β, α)-fuzzy bi-ideals, and (β, α)-fuzzy interior ideals, where α, β ϵ{ ϵγ, qσ, ϵγ ∨ qσ, ϵγ ∧ qσ} and α, β ϵ{ϵγ, qσ, ϵγ ∨ qσ, ϵγ ∧ qσ}, with α ≠ ϵγ ∧ qσ and β ≠ ϵγ ∧ qσ. An important milestone achieved by this paper is providing the connection between classical algebraic structures (ordinary bi-ideals, interior ideals) and new types of fuzzy algebraic structures [(ϵγ, ϵγ ∨ qσ)-fuzzy bi-ideals, (ϵγ, ϵγ ∨ qσ)-fuzzy interior ideals]. Special attention is given to (ϵγ, ϵγ ∨ qσ)-fuzzy bi-ideals and (ϵγ, ϵγ ∨ qσ ) -fuzzy bi-ideals.