Picture Fuzzy Ideals in Semigroups under the Type-3 Order: Bi-Ideals, Quasi-Ideals, and Interior Ideals

This paper studies picture fuzzy subsets of semigroups under the order (D^*,≤_3 ). We first clarify the admissible domain D^*={(φ,ψ,ω)∈[0,1]^3:0≤φ+ψ+ω≤1} and then redefine basic operations (intersection, product, Cartesian product, image, inverse image, and characteristic function) so that all results remain inside the picture fuzzy framework. Based on these operations, we define picture fuzzy subsemigroups, bi-ideals, quasi-ideals, and interior ideals, and we prove their fundamental closure and characterization properties. We also provide counterexamples showing why some earlier formulations can fail to produce valid picture fuzzy sets. Finally, we show that the corresponding ideal families form complete lattices and that these structures are preserved under standard homomorphic constructions.