Soft Intersection Bi-interior Ideals of Semigroups

Generalizing the ideals of an algebraic structure has shown to be both beneficial and interesting for mathematicians. In this context, the idea of the bi-interior ideal was introduced as a generalization of the bi-ideal and interior ideal of a semigroup. By introducing "soft intersection (₷-int) bi-interior (ᙝᏆ) ideals of semigroups", we introduce a framework integrating semigroup theory with soft set theory in this study. Finding the relationships between ₷-int ᙝᏆ-ideals and other specific kinds of ₷-int ideals of a semigroup is the main aim of this study. Our results show that an ₷-int ᙝᏆ-ideal is an ₷-int subsemigroup of a soft simple* semigroup, and that an ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal is an ₷-int ᙝᏆ-ideal; in other words, the ₷-int ᙝᏆ-ideal is a generalization of the ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal, however, we provide counterexamples demonstrating that the converses do not always hold. We demonstrate that the semigroup should be a soft simple* semigroup in order to satisfy the converses. Our key theorem, which states that if a nonempty subset of a semigroup is a ᙝᏆ-ideal, then its soft characteristic function is an ₷-int ᙝᏆ-ideal, and vice versa, enables us to bridge the gap between semigroup theory and soft set theory. Using this theorem, we show how this idea relates to the existing algebraic structures in classical semigroup theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of soft set operations supporting our assertions with illuminating examples.