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

This thesis is a study in the field of soft set theory, which was proposed by Molodstov to model problems involving uncertainty. In this study, in order to contribute to soft set theory, we introduce three different soft intersection (almost) ideals of semigroups, examine the reflections of transitions between ideals in classical semigroup theory, and present the findings in a systematic manner. This thesis consists of six chapters. In the first chapter, literature related to semigroups, soft sets, semigroups, and the ideals of semigroups are reviewed. The second chapter covers the fundamental concepts related to the semigroups, the ideals and almost ideals of semigroups, soft sets, and soft intersection (almost) ideals of semigroups used in this thesis. In the third chapter, the concept of soft intersection bi-quasi-interior ideals of semigroups are defined, illustrated with examples, and the significant relationships of these soft intersection ideals between other soft interection ideals of semigroup are explored. It is proven with theorems that every soft intersection bi-ideal, soft intersection interior ideal, soft intersection left (right) ideal, soft intersection quasi-ideal, soft intersection bi-interior ideal, soft intersection left (right) bi-quasi ideal, and the soft intersection left (right) quasi-interior ideal is a soft intersection bi-quasi-interior ideal of a semigroup, and that the converses may hold under certain conditions. Additionally, it is shown that the soft characteristic function of a subgroup of a semigroup, which is a bi-quasi-interior ideal, is a soft intersection bi-quasi-interior ideal of a semigroup and vice versa. Thus, an important connection between classical semigroup theory and soft set theory is established. In the fourth chapter, the concept of soft intersection almost interior ideals of semigroups are defined. The fifth chapter introduces and illustrates the soft intersection almost bi-quasi-interior ideals of semigroups. It is demonstrated that the soft characteristic functions of a nonempty subset of a semigroup that is almost interior (bi-quasi-interior) ideal is a soft intersection almost interior (bi-quasi-interior) ideal of a semigroup, respectively. Furthermore, the definitions and theorems regarding the minimality, primeness, semiprimeness, and strongly primeness of an almost ideal of a semigroup are generalized to soft intersection almost ideals of semigroups. The sixth and final chapter discusses the results obtained and the significance of the findings of this study.