Soft Intersection Bi-quasi Ideals of Semigroup

Abstract

Mathematicians find it valuable to extend the concept of ideals within algebraic structures. The bi-quasi (ƁԚ) ideal was introduced as a broader version of quasi-ideal, bi-ideal, and left (right) ideals in semigroups. This paper applies this concept to soft set theory and semigroups, introducing the "Soft intersection (S-int) ƁԚ ideal." The goal is to explore the relationships between S-int ƁԚ ideals and other types of S-int ideals in semigroups. It is shown that every S-int bi-ideal, S-int ideal, S-int quasi-ideal, and S-int interior ideal of an idempotent soft set are S-int ƁԚ ideals. Counterexamples demonstrate that the reverse is not always true unless the semigroup is simple* or regular. For soft simple* semigroups, the S-int ƁԚ ideal coincides with the S-int bi-ideal, S-int left (right) ideal, and S-int quasi-ideal. The main theorem shows that if a subsemigroup of a semigroup is a ƁԚ ideal, its soft characteristic function is an S-int ƁԚ ideal, and vice versa. This connects semigroup theory with soft set theory. The paper also discusses how this concept integrates into classical semigroup structures, providing characterizations and analysis using soft set operations, soft image, and soft inverse image, supported by examples.

Keywords

• Soft set
• semigroup
• bi-quasi ideals
• soft intersection bi-quasi ideals
• simple* semigroups

Soft Intersection Bi-quasi Ideals of Semigroup

Öz

Matematikçiler, cebirsel yapılardaki ideal kavramını genişletmeyi değerli bulmaktadır. Bi-quasi (ƁԚ) ideal, yarıgruplarda quasi-ideal, bi-ideal ve sol (sağ) idealin daha geniş bir versiyonu olarak tanıtılmıştır. Bu makale, bu kavramı esnek küme teorisi ve yarıgruplara uygulayarak "Esnek kesişimsel (EK) ƁԚ ideali" tanıtmaktadır. Amaç, EK ƁԚ idealleri ile diğer EK ideal türleri arasındaki ilişkileri incelemektir. Bir idempotent esnek küme için her EK-bi-ideal, EK-ideal, EK-quasi-ideal ve EK-iç idealin aynı zamanda bir EK*ƁԚ ideal olduğu gösterilmiştir. Ancak, tersinin her zaman geçerli olmadığı, yalnızca yarıgrubun basit* veya regüler olduğunda sağlandığı aksine örneklerle gösterilmiştir. Esnek basit* yarıgruplarda, EK-ƁԚ idealin EK-bi-ideal, EK-sol (sağ) ideal ve EK- quasi ideal ile çakıştığı kanıtlanmıştır. Ana teorem, bir yarıgrubun alt yarıgrubu bir ƁԚ ideal ise, onun esnek karakteristik fonksiyonunun bir EK-ƁԚ ideal olduğunu ve bunun tersinin de geçerli olduğunu göstermektedir. Bu sonuç, yarıgrup teorisi ile esnek küme teorisi arasındaki bağlantıyı kurmaktadır. Ayrıca, bu kavramın klasik yarıgrup yapılarıyla nasıl bütünleştiği tartışılmakta ve esnek küme işlemleri, esnek görüntü ve esnek ters görüntü kullanılarak çeşitli karakterizasyonlar ve analizler yapılmıştır. Bulgular örneklerle desteklenmiştir.

Anahtar Kelimeler

• esnek küme
• yarıgrup
• bi-quasi idealler
• esnek kesişimsel bi-quasi idealler
• basit* yarıgruplar

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