Extended Bipolar Intuitionistic Fuzzy Ideals Framework Through Level Sets and Its Characterization via Regular Ordered Γ-Semigroups

This paper proposes an extended framework for bipolar anti-intuitionistic fuzzy ideals within the context of ordered Γ-semigroups. We introduce and investigate the (δ, τ)-bipolar anti-intuitionistic fuzzy subsemigroups (BPAIFSS), including their associated left ideals, right ideals, ideals, and bi-ideals. These structures generalize existing fuzzy ideal notions by incorporating dual-valued membership and non-membership functions with flexible threshold control. Using level set analysis, we characterize the algebraic properties of these fuzzy ideals and establish their role in determining the regularity of ordered Γ-semigroups. Illustrative examples are provided to validate and demonstrate the applicability of the theoretical results.