Soft Intersection Weak-interior Ideals of Semigroups

Authors : Aslıhan Sezgin, Aleyna İlgin, M Murali

Pages : 470-483

View : 59 | Download : 59

Publication Date : 2025-08-30

Article Type : Research Paper

Abstract :Abstract: The idea of generalization of ideals of the algebraic structures has always shown to be interesting for mathematicians. Within this framework, the notion of a weak-interior ideal has been introduced as a generalization of quasi-ideals, interior ideals, (left/right) ideals in the theory of semigroups. In the present work, we extend this concept into the framework of soft set theory applied to semigroups, and introduce a novel type of the soft intersection (ᵴ-intersection) ideal called soft intersection (ᵴ -intersection) weak-interior ideal. The main aim of this study is to investigate the reliations of ᵴ-intersection weak-interior ideals with other types of ᵴ-intersection ideals within semigroups. Our results establish that every ᵴ-intersection weak-interior ideal constitutes an ᵴ-intersection subsemigroup within a regular semigroup. Moreover, every ᵴ-intersection left (right) ideal is an ᵴ-intersection left (right) weak-interior ideal, and every ᵴ-intersection interior ideal is an ᵴ-intersection weak-interior ideal. Consequently, the concept of an ᵴ-intersection weak-interior ideal is a generalization of both ᵴ-intersection ideals and ᵴ-intersection interior ideals. However, the converses do not hold in general with counterexamples. To satisfy the converses, semigroup should be group or the ᵴ-intersection weak-interior ideal should be idempotent. Furthermore, we prove that ᵴ-intersection bi-ideals and ᵴ-intersection quasi-ideals coincide with ᵴ-intersection weak-interior ideals within the framework of group structures. Our key theorem, which shows that if a subsemigroup of a semigroup is a weak-interior ideal, then its soft characteristic function is an ᵴ-intersection weak-interior ideal, and vice versa, allows us to build a bridge between semigroup theory and soft set theory.
Keywords : Esnek kümeler, Yarıgruplar, Zayıf-iç idealler, Esnek kesişimsel zayıf-iç idealler, Basit yarıgruplar

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