Soft Union Bi-quasi Ideals of Semigroup

Mathematicians attach importance to extending ideals in algebraic structures. The concept of bi-quasi (ƁԚ) ideal was introduced as a generalization 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 notion of "Soft union (S-uni) ƁԚ ideal." The aim of this paper is to explore the relationships between S-uni ƁԚ ideals and other types of S-uni ideals in semigroups. It is shown that every S-uni bi-ideal, S-uni ideal, S-uni quasi-ideal, and S-uni interior ideal of an idempotent soft set are S-uni ƁԚ ideals. Counterexamples demonstrate that the converses are not always true unless the semigroup is special soft simple or regular. For special soft simple semigroups, the S-uni ƁԚ ideal coincides with the S-uni bi-ideal, S-uni left (right) ideal, and S-uni quasi-ideal. Additionally, we provide conceptual definitions and analyses of the new concept in the context of soft set operations, supporting our claims with clear examples.