Soft Intersection Almost Bi-quasi Ideals of Semigroups

Abstract

The concept of bi-quasi ideal generalizes the notions of bi-ideals and quasi-ideals in a semigroup; similarly, the soft intersection bi-quasi ideal generalizes the concepts of soft intersection bi-ideals and soft intersection quasi-ideals in a semigroup. In this paper, we introduce the concept of  soft intersection almost bi-quasi ideal and its generalized concept, soft intersection weakly almost bi-quasi ideals, in a semigroup. In contrast to the soft intersection ideal theory, we demonstrate that every soft intersection almost bi-quasi ideal is also a soft intersection almost ideal and a soft intersection almost bi-ideal. Additionally, we show that every idempotent soft intersection almost bi-quasi ideal is a soft intersection almost subsemigroup, a soft intersection almost weak interior ideal, a soft intersection almost tri-ideal, and a soft intersection almost tri-bi-ideal. Furthermore, we derive several interesting relationships regarding minimality, primeness, semiprimeness, and strongly primeness between almost bi-quasi ideals and soft intersection almost bi-quasi ideals with the proven theorem stating that if a nonempty set  is an almost bi-quasi ideal, then its soft characteristic function is also a soft intersection almost bi-quasi ideal, and vice versa.