Soft intersection almost bi-quasi-interior ideals of semigroups - Aslıhan Sezgin; Zeynep BAŞ; Aleyna İlgin (Journal of Fuzzy Extension and Applications)
Generalizing the ideals of an algebraic structure has shown to be both exciting and valuable to mathematicians. In this context, the concept of Bi-Quasi-Interior Ideal (BQI-ideal) was proposed as a generalization of interior ideal, quasi-ideal, bi-ideal, bi-quasi ideal, and bi-interior ideal in a semigroup in a manner similar to how the notion of soft intersection BQI-ideal was presented to generalize the soft intersection interior ideal, quasi-ideal, bi-ideal, bi-quasi ideal, and bi-interior ideal in a semigroup. In this study, we introduce new types of soft intersection almost ideals called "soft intersection (weakly) almost BQI-ideal" and discuss the concepts in terms of their algebraic structures, providing some characterizations regarding the concepts. To explain the essential properties of these ideals, we employ algebraic methods. The primary objective of this paper is to establish the relationship between soft intersection (weakly) almost BQI-ideals and other certain types of almost ideals in semigroups and soft intersection ideals. It is obtained that a nonnull soft intersection BQI-ideal is a soft intersection almost BQI-ideal and that a soft intersection almost BQI-ideal is a soft intersection weakly almost BQI-ideal; however, the counterparts are not valid with counterexamples. Additionally, we obtain that any idempotent soft intersection almost BQI-ideal is a soft intersection almost subsemigroup. With our obtained crucial theorem that states that if a nonempty set of a semigroup is an almost BQI-ideal, then its soft characteristic function is a soft intersection almost BQI-ideal, and vice versa, we achieve to construct a bridge between semigroup theory and soft set theory. With this vital theorem, several interesting relationships between certain types of almost BQI-ideals of semigroups, such as minimal, prime, semiprime, and strongly prime almost BQI-ideals and certain types of soft intersection almost BQI-ideals, are derived.  Besides, we support our assertions with illustrative and concrete examples.