On the Structure of Bipolar Fuzzy Bi-Quasi Ideals in Γ-Semirings

In this paper, we introduce the notion of bipolar fuzzy bi-quasi ideals in [Formula: see text]-semirings and investigate their fundamental properties. The relationships between bipolar fuzzy left bi-quasi ideals, bipolar fuzzy right bi-quasi ideals, bipolar fuzzy ideals, and bipolar fuzzy quasi ideals are examined in detail. Several characterization results are obtained by means of level subsets and bipolar fuzzy characteristic functions. Conditions under which bipolar fuzzy bi-quasi ideals coincide with other generalized bipolar fuzzy ideals are established. We also study the behavior of these ideals under intersection and homomorphic inverse images. Further, regular [Formula: see text]-semirings are characterized in terms of bipolar fuzzy bi-quasi ideals. Illustrative examples are provided to support the theoretical results. The results obtained extend existing studies on bipolar fuzzy generalized ideals and contribute to the structural theory of [Formula: see text]-semirings.