Some Innovative Types of Fuzzy Ideals in AG-Groupoids

Abstract

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, (β̄, ᾱ)-fuzzy bi-ideals, and (β̄, ᾱ)-fuzzy interior ideals, where α, β∈{∈γ, qδ, ∈γ∨qδ, ∈γ∧qδ} and ᾱ, β̄∈{⋶γ, q̄δ, ⋶γ∨q̄δ, ⋶γ∧q̄δ}, with α≠∈γ∧qδ and β̄≠⋶γ∧q̄δ. 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 [(∈γ, ∈γ∨qδ)-fuzzy bi-ideals, (∈γ, ∈γ∨qδ)-fuzzy interior ideals]. Special attention is given to (∈γ, ∈γ∨qδ)-fuzzy bi-ideals and (⋶γ, ⋶γ∨q̄δ) -fuzzy bi-ideals.