Soft Intersection Almost Tri-ideals of Semigroups

The notions of left (right) ideal and quasi-ideal of semigroup are generalized by the concept of tri-ideal. Likewise, the concepts of soft intersection left (right) ideal and soft intersection quasi-ideal of semigroups are generalized by the soft intersection tri-ideal. In this paper, we present and study the concept of soft intersection almost tri-ideal ideal as a further generalization of nonnull soft intersection tri-ideal. It is demonstrated that every idempotent soft intersection almost bi-ideal is a soft intersection almost tri-ideal, and vice versa. We also obtained that an idempotent soft intersection almost left (or right) tri-ideal coincides with the soft intersection almost tri-ideal. It is also shown that every idempotent soft intersection almost tri-ideal is a soft intersection almost subsemigroup. With the noteworthy result that if a nonempty subset of a semigroup is an almost tri-ideal, then its soft characteristic function is also a soft intersection almost tri-ideal, and vice versa, a number of intriguing relationships regarding minimality, primeness, semiprimeness, and strongly primeness between almost tri-ideals and soft intersection almost tri-ideals are derived.