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

Abstract

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.

Keywords

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

References

1. Good RA, Hughes DR. Associated groups for a semigroup. Bull Amer Math Soc. 1952;58(6):624-625.
2. Steinfeld O. Uher die quasi ideals. Von halbgruppend Publ Math Debrecen. 1956;4:262-275.
3. Lajos S. (m;k;n)-ideals in semigroups. Notes on Semigroups II, Karl Marx Univ. Econ., Dept. Math. Budapest. 1976;(1):12-19.
4. Szasz G. Interior ideals in semigroups. Notes on semigroups IV, Karl Marx Univ. Econ., Dept. Math. Budapest. 1977;(5):1-7.
5. Szasz G. Remark on interior ideals of semigroups. Studia Scient. Math. Hung. 1981;(16):61-63.
6. Rao MMK. Bi-interior ideals of semigroups. Discuss Math Gen Algebra Appl. 2018;38(1) 69-78.
7. Rao MK. A study of a generalization of bi-ideal, quasi-ideal and interior ideal of semigroup. Mathematica Moravica. 2018;22(2):103-115.
8. Rao MK. Left bi-quasi ideals of semigroups. Southeast Asian Bull Math. 2020;44:369-376.

9. Rao MMK. Quasi-interior ideals and weak-interior ideals. Asia Pac Journal Mat. 2020;7:7-21.
10. Baupradist S, Chemat B, Palanivel K, Chinram R. Essential ideals and essential fuzzy ideals in semigroups. J Discrete Math Sci Cryptogr. 2021;24(1): 223-233.
11. Grošek O, Satko L. A new notion in the theory of semigroup. Semigroup Forum. 1980;20(1):233-240.
12. Bogdanovic S. Semigroups in which some bi-ideal is a group. Univ u Novom Sadu Zb Rad Prirod Mat Fak Ser Mat. 1981;11:261-266.
13. Wattanatripop K, Chinram R, Changphas T. Quasi-A-ideals and fuzzy A-ideals in semigroups. J Discrete Math Sci Cryptogr. 2018;21(5):1131-1138.
14. Kaopusek N, Kaewnoi T, Chinram R. On almost interior ideals and weakly almost interior ideals of semigroups. J Discrete Math Sci Cryptogr. 2020;23(3):773-778.
15. Iampan A, Chinram R, Petchkaew P. A note on almost subsemigroups of semigroups. Int J Math Comput Sci. 2021;16:1623-1629.
16. Chinram R, Nakkhasen W. Almost bi-quasi-interior ideals and fuzzy almost bi-quasi-interior ideals of semigroups. J Math Comput Sci. 2022;26:128-136.
17. Gaketem T. Almost bi-interior ideal in semigroups and their fuzzifications. Eur J Pure Appl Math. 2022;15(1):281-289.
18. Gaketem T, Chinram R. Almost bi-quasi ideals and their fuzzifications in semigroups. Ann Univ Craiova Math Comput Sci Ser. 2023;50(2):342-352.
19. Wattanatripop K, Chinram R, Changphas T. Fuzzy almost bi-ideals in semigroups. Int J Math Comput Sci. 2018;13(1):51-58.
20. Krailoet W, Simuen A, Chinram R, Petchkaew P. A note on fuzzy almost interior ideals in semigroups. Int J Math Comput Sci. 2021;16(2):803-808.
21. Molodtsov D. Soft set theory-first results. Comput Math Appl. 1999;37(1):19-31.
22. Maji PK, Biswas R, Roy AR. Soft set theory. Comput Math Appl. 2003;45(1):555-562.
23. Pei D, Miao D. From soft sets to information systems. In: Proceedings of Granular Computing. IEEE 2005;2 617–621.
24. Ali MI, Feng F, Liu X, Min WK, Shabir M. On some new operations in soft set theory. Comput Math Appl. 2009;57(9):1547–1553.
25. Sezgin A, Dagtoros K. Complementary soft binary piecewise symmetric difference operation: A novel soft set operation. Scientific Journal of Mehmet Akif Ersoy University. 2023;6(2):1457-1467.
26. Sezgin A, Sarıalioğlu M. A new soft set operation: complementary soft binary piecewise theta (𝛉) operation. Journal of Kadirli Faculty of Applied Sciences. 2024;4(2):325-357.
27. Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl. 2011;61(9):2647-2654.
28. Sezgin A, Çağman N, Atagün AO, Aybek FN. A. Complemental binary operations of sets and their application to group theory. Matrix Science Mathematic, 2023;7(2):114-121.
29. Stojanović NS. A new operation on soft sets: extended symmetric difference of soft sets. Military Technical Courier. 2021;69(4):779-791.
30. Sezgin A, Yavuz E. Soft binary piecewise plus operation: A new type of operation for soft sets. Uncertainty Discourse and Applications. 2024;1(1):79-100.
31. Sezgin A, Çalışıcı H. A comprehensive study on soft binary piecewise difference operation. Eskişeh Tek Univ Bilim Teknol Derg Teor Bilim. 2024;12:32-54.
32. Sezgin A, Atagün AO, Çağman N. A complete study on and-product of soft sets. Sigma J Eng Nat Sci. 2025;43(1):1-14.
33. Sezgin A, Aybek FN, Güngör NB. A new soft set operation: complementary soft binary piecewise union operation. Acta Informatica Malaysia. 2023;7(1):38-53.
34. Sezgin A, Sarıalioğlu M. Complementary extended gamma operation: A new soft set operation. Natural and Applied Sciences Journal. 2024;7(1):15–44.
35. Sezgin A, Yavuz E. A new soft set operation: complementary soft binary piecewise lambda operation. Sinop University Journal of Natural Sciences. 2023;8(5):101-133.
36. Sezgin A, Yavuz E. A new soft set operation: soft binary piecewise symmetric difference operation. Necmettin Erbakan University Journal of Science and Engineering. 2023;5(2):189-208.
37. Sezgin A, Çağman N. A new soft set operation: complementary soft binary piecewise difference operation. Osmaniye Korkut Ata Üniv Fen Biliml Derg. 2024;7(1):58-94.
38. Çağman N, Enginoğlu S. Soft set theory and uni-int decision making. Eur J Oper Res. 2010;207(2):848-855.
39. Çağman N, Çıtak F, Aktaş H. Soft int-group and its applications to group theory. Neural Comput Appl. 2012;21:151-158.
40. Sezer AS, Çağman N, Atagün AO, Ali MI, Türkmen E. Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I. Filomat. 2015;29(5):917-946.
41. Sezer AS, Çağman N, Atagün AO. Soft intersection interior ideals, quasi-ideals and generalized bi-ideals: a new approach to semigroup theory II. J Mult-Valued Log Soft Comput. 2014;23 (1–2) 161-207.
42. Sezgin A, Orbay M. Analysis of semigroups with soft intersection ideals. Acta Univ Sapientiae Math. 2022;14(1):166-210.
43. Sezgin A, İlgin A. Soft intersection almost ideals of semigroups. J Innovative Eng Nat Sci. 2024;4(2):466-481.
44. Sezgin A, İlgin A. Soft intersection almost bi-interior ideals of semigroups. J Nat Appl Sci Pak. 2024;6(1):1619-1638.
45. Sezgin A, İlgin A. Soft intersection almost bi-quasi ideals of semigroups. Soft Computing Fusion with Applications. 2024;1(1):27-42.
46. Sezgin A, İlgin A. Soft intersection almost weak-interior ideals of semigroups. Journal of Natural Sciences and Mathematics of UT. 2024;9(17-18):372-385.
47. Sezgin A, Onur B. Soft intersection almost bi-ideals of semigroups. Syst Anal. 2024;2(1):95-105.
48. Sezgin A, Kocakaya FZ, İlgin A. Soft intersection almost quasi-interior ideals of semigroups. Eskişeh Tek Univ Bilim Teknol Derg Teor Bilim. 2024;12(2):81-99.
49. Sezgin A, İlgin A. Soft intersection almost subsemigroups of semigroups. Int J Math Phys. 2024;15(1):13-20.
50. Sezgin A, Baş ZH, İlgin A. Soft intersection almost bi quasi-interior ideals of semigroups. J Fuzzy Ext Appl. 2025;6(1),43-58.
51. Sezgin A, Onur B, İlgin A. Soft intersection almost tri-ideals of semigroups. SciNexuses. 2024;1:126-138.
52. Sezgin A, Ilgin A, Atagun AO. Soft intersection almost tri-bi-ideals of semigroups. Sci Technol Asia. 2024;29(4):1-13.
53. Sezgin A, Kocakaya FZ. Soft intersection almost quasi-ideals of semigroups. Songklanakarin J Sci Technol. 2025;47(2): in press.
54. Sezgin A, Baş ZH. Soft-int almost interior ideals for semigroups. Information Information Sci Appl. 2024;4:25-36.
55. Atagün AO, Kamacı H, Taştekin İ, Sezgin A. P-properties in near-rings. J. Math. Fundam. Sci. 2019;51(2):333-350.
56. Jana C, Pal M, Karaaslan F, Sezgı̇n A. (α, β)-Soft intersectional rings and ideals with their applications. New Math Nat Comput. 2019;15(02):333-350.
57. Atagün AO, Sezgin A. More on prime, maximal and principal soft ideals of soft rings. New Math Nat Comput. 2022;18(1):195-207.
58. Atagün AO, Sezgin A. Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences (AMIS). 2017;11(1):105-113.
59. Manikantan T, Ramasany P, Sezgin A. Soft quasi-ideals of soft near-rings. Sigma J. Eng. Nat. Sci. 2023;41(3):565-574.
60. Atagün AO, Sezgin A. A new view to near-ring theory: Soft near-rings. South East Asian J. of Math. & Math. Sci. 2013;14(3):1-14.
61. Atagün AO, Sezgin A. Soft subnear-rings, soft ideals and soft N-subgroups of near-rings. Math Sci Letters. 2018;7:37-42.
62. Sezgin A. A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat. 2018;32(8):2995-3030.
63. Sezer AS, Atagün AO, Çağman N. A new view to N-group theory: soft N-groups. Fasc Math. 2013;51:123-140.
64. Sezer AS, Atagün AO, Çağman N. N-group SI-action and its applications to N-group theory. Fasc Math. 2014;52:139-153.
65. Gulistan M, Feng F, Khan M, Sezgin A. Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets, Mathematics. 2018;6:293.
66. Khan A, Izhar M, Sezgin A. Characterizations of Abel Grassmann’s groupoids by the properties of double-framed soft ideals. Int J Anal Appl. 2017:15(1):62-74.
67. Atagün AO, Sezer AS. Soft sets, soft semimodules and soft substructures of semimodules. Math Sci Lett. 2015;4(3):235-242.
68. Sezgin A, İlgin A. Soft Intersection bi-interior ideals of semigroups. J Innov Sci Eng. 2025; 9(1): in press.
69. Rao MMK. Bi-quasi ideals and fuzzy bi-quasi ideals of semigroups. Bull. Int. Math. Virtual Inst. 2017;7(2):231-242.
70. Rao MMK. Left bi-quasi ideals of semirings. Bull. Int. Math. Virtual Inst. 2018;8(1):45-53.
71. Rao MMK. Fuzzy left and right bi-quasi ideals of semiring. Bull. Int. Math. Virtual Inst. 2018;8(3):449-460.
72. Rao MMK, Venkateswarlu B, Rafi N. Left bi-quasi ideals of Γ-semirings. Asia Pac J Math. 2017;4(2):144-153.