Soft Intersection Bi-quasi Ideals of Semigroup

Mathematicians find it valuable to extend the concept of ideals within algebraic structures. The bi-quasi (ƁԚ) ideal was introduced as a broader version of quasi-ideal, bi-ideal, and left (right) ideals in semigroups. This paper applies this concept to soft set theory and semigroups, introducing the "Soft intersection (S-int) ƁԚ ideal." The goal is to explore the relationships between S-int ƁԚ ideals and other types of S-int ideals in semigroups. It is shown that every S-int bi-ideal, S-int ideal, S-int quasi-ideal, and S-int interior ideal of an idempotent soft set are S-int ƁԚ ideals. Counterexamples demonstrate that the reverse is not always true unless the semigroup is simple* or regular. For soft simple* semigroups, the S-int ƁԚ ideal coincides with the S-int bi-ideal, S-int left (right) ideal, and S-int quasi-ideal. The main theorem shows that if a subsemigroup of a semigroup is a ƁԚ ideal, its soft characteristic function is an S-int ƁԚ ideal, and vice versa. This connects semigroup theory with soft set theory. The paper also discusses how this concept integrates into classical semigroup structures, providing characterizations and analysis using soft set operations, soft image, and soft inverse image, supported by examples.