Soft intersection (almost) quasi-interior ideals and soft intersection almost quasi-ideals of semigroups

The new method used in modeling uncertainty, called fuzzy set theory, was first introduced by Molodtsov in 1999, and the theory has been applied to many mathematical fields. In this study, some new soft intersection ideals and soft intersection almost ideals of semigroups have been proposed for the purpose of generalizing some certain soft intersection ideals of semigroups. In this regard, "Soft Intersection Almost Quasi-ideal" and "Soft Intersection (Almost) Quasi-interior Ideal" of a semigroup have been presented, their properties together with their relationships with other soft intersection (almost) ideals have been explored. This study consists of six chapters. In the first chapter, studies in the literature related to soft sets, soft set operations, semigroups, and ideals of semigroups are discussed. The second chapter introduces the definitions, theorems, and concepts used in this thesis. In the third chapter, the concept of soft intersection quasi-interior ideal of a semigroup is proposed, its properties are provided, and its relationships with other soft intersection ideals of semigroups are given and illustrated with examples. It is also proven that a nonempty subset of a semigroup is a quasi-interior ideal if and only if its soft chracteristic function is a soft intersection quasi-interior ideal. In the fourth chapter, the concepts of soft intersection almost quasi-ideal and its generalization, soft intersection weak almost quasi-ideal of a semigroup; in the fifth chapter, soft intersection almost quasi-interior ideal and its generalization, soft intersection weak almost quasi-interior ideal of a semigorup are provided and their relationships with other soft intersection almost ideals are obtained. Additionally, the relationships between minimality, primeness, semiprimeness and strongly primeness between almost quasi (quasi-interior) ideals and soft intersection almost quasi (quasi-interior) ideals of a semigroup are demonstrated, and some characterizations of these structures are obtained. Thus, the relationships between classical semigroup theory and soft set theory are demonstrated in this study. The final chapter presents the conclusions obtained from this thesis, the significance of the results, and the suggestions for future work.