Soft Intersection Weak-interior Ideals of Semigroups

Year 2025, Volume: 41 Issue: 2, 470 - 483, 30.08.2025

Aslıhan Sezgin , Aleyna İlgin , M Murali

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

Soft sets , Semigroups , Weak-interior ideals , Soft intersection weak-interior ideals , Simple semigroups

Yarıgrupların Esnek Kesişimsel Zayıf-iç İdealleri

Abstract

Öz: Cebirsel yapıların ideallerin genelleştirilmesi matematikçilere her zaman ilginç gelmiştir. Bu çerçevede, zayıf-iç ideal kavramı, yarıgruplar teorisindeki quasi-idealleri, iç idealleri, (sol/sağ) idealleri kapsayan bir genelleme olarak tanıtılmıştır. Mevcut çalışmada, bu yapıyı yarıgruplara uygulanan esnek küme teorisi çerçevesinde genişletip, esnek kesişim (EK) zayıf-iç ideal adı verilen yarıgrupların yeni bir esnek kesişim (EK) ideal türünü tanıtıyoruz. Bu çalışmanın temel amacı, EK-zayıf-iç ideallerinin yarıgruplar içindeki diğer EK-ideal türleriyle ilişkilerini araştırmaktır. Sonuçlarımız, regüler bir yarıgrubun her EK-zayıf-iç idealinin bir EK-alt yarıgrup oluşturduğunu ortaya koymaktadır. Ayrıca, her EK-sol (sağ) ideal bir EK-sol (sağ) zayıf-iç idealdir ve her EK-iç ideal bir EK-zayıf-iç idealdir. Sonuç olarak, EK-zayıf-iç ideal kavramı, EK-ideallerin hem de EK-iç ideallerin bir genellemesidir. Ancak, karşıtları genel olarak geçerli değildir ve bunu gösteren açık karşıt örnekler sunulmuştur. Karşıtlarının sağlanması için, yarıgrubub grup olması veya EK-zayıf-iç idealin idempotent olması gerekmektedir. Ayrıca, EK-bi-ideallerin ve EK-quasi-ideallerin grup yapıları çerçevesinde EK-zayıf-iç ideallerle çakıştığı kanıtlanmıştır. Bir yarı grubun alt yarıgrubu zayıf-iç ideal ise, esnek karakteristik fonksiyonunun EK-zayıf-iç ideal olduğunu ve karşıtının da doğru olduğunu gösteren temel teoremimiz, yarıgrup teorisi ile esnek küme teorisi arasında bir köprü kurmamızı sağlamıştır.

Keywords

Esnek kümeler , Yarıgruplar , Zayıf-iç idealler , Esnek kesişimsel zayıf-iç idealler , Basit yarıgruplar

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