Soft intersection bi-quasi ideals, soft intersection almost bi-ideals and tri-ideals of semigroups

The theory of soft sets has been recognized as a new mathematical approach since it was introduced by Molodtsov in 1999, and it has a wide range of applications in algebraic contexts. Ideals are essential for the advanced study of algebraic structures and their applications. To advance research in this field, it is necessary to develop and generalize the ideals within algebraic structures. In this thesis, new types of soft intersection (almost) ideals of semigroups, named "Soft Intersection Bi-quasi Ideal," "Soft Intersection Almost Bi-ideal," and "Soft Intersection Almost Tri-ideal," are introduced to contribute to soft set theory, their properties are expressed, and their relationships with other soft intersection (almost) ideals of semigroups are examined. To establish a bridge between soft set theory and classical semigroup theory, some important connections have been obtained. This study consists of six chapters. The first chapter discusses the existing literature on semigroups, soft sets, soft intersection ideals, and soft intersection almost ideals of semigroups. The second chapter presents the basic definitions, properties, and theorems that are used in this thesis. The third chapter introduces the soft intersection bi-quasi ideal of semigroups, provides its examples, and examines its relationship with certain soft intersection ideals of semigroups. Under necessary conditions, it is shown that a soft intersection ideal (bi-ideal, quasi-ideal, interior ideal) is a soft intersection bi-quasi ideal of a semigroup. Counterexamples are presented to demonstrate that the converse of these statements does not always hold, and it is proven that the semigroup has to be a soft simple* or regular semigroup for the converse to be valid. In the fourth and fifth chapters, the soft intersection almost bi-ideal and soft intersection almost tri-ideal of a semigroup are respectively introduced, their examples are provided, and their relationships with other soft intersection (almost) ideals are analyzed. Under sufficient conditions, it is shown that a soft intersection bi-ideal (tri-ideal) is an almost bi-ideal (tri-ideal), and that an almost bi-ideal (tri-ideal) is a soft intersection almost subsemigroup. The final chapter provides a detailed discussion of the results obtained from this study and the significance of these results.