Soft Intersection Bi-quasi-interior Ideals of Semigroups

Abstract

In this paper, we introduce the concept of the soft intersection (S-int) bi-quasi-interior (₿QĪ) ideal of semigroups and provide an equivalent definition. The relationships between S-int ideals and S-int ₿QĪ ideals are established. We prove that every S-int bi-ideal, left ideal, right ideal, interior ideal, quasi-ideal, bi-interior ideal, left/right bi-quasi-ideal, and left/right quasi-interior ideal is also an S-int ₿QĪ ideal. Counterexamples are given to show that the converses do not hold, and we demonstrate that additional conditions, such as regularity or right/left simplicity, are required for the converses. We also show that if a subsemigroup of a semigroup is a ₿QĪ ideal, then its soft characteristic function is an S-int ₿QĪ ideal, and the converse holds as well. Thus, this work establishes an important connection between classical semigroup theory and soft set theory. Furthermore, we show that finite soft AND-products, Cartesian products, and intersections of S-int ₿QĪ ideals remain S-int ₿QĪ ideals, whereas finite soft OR-products and unions do not. This study provides a broad conceptual characterization and analysis of S-int ₿QĪ ideals.