Araştırma Makalesi

Yıl 2024,
Cilt: 12 Sayı: 2, 81 - 99, 29.08.2024
### Öz

### Anahtar Kelimeler

### Kaynakça

Similar to how the quasi-interior ideal generalizes the ideal and interior ideal of a semigroup, the concept of soft intersection quasi-interior ideal generalizes the idea of soft intersection ideal and soft intersection interior ideal of a semigroup. In this study, we provide the notion of soft intersection almost quasi-interior ideal as well as the soft intersection weakly almost quasi-interior ideal in a semigroup. We show that any nonnull soft intersection quasi-interior ideal is a soft intersection almost quasi-interior ideal; and soft intersection almost quasi-interior ideal is a soft intersection weakly almost quasi-interior ideal, but the converses are not true. We further demonstrate that any idempotent soft intersection almost quasi-interior ideal is a soft intersection almost subsemigroup. With the established theorem that states that if a nonempty set A is almost quasi-interior ideal, then its soft characteristic function is a soft intersection almost quasi-interior ideal, and vice versa, we are also able to derive several intriguing relationships concerning minimality, primeness, semiprimeness, and strongly primeness between almost quasi-interior ideals, and soft intersection almost quasi-interior ideals.

Esnek küme Yarıgrup (Hemen hemen) quasi-iç idealler Esnek kesişimsel (hemen hemen) quasi-iç idealler

- [1] Good RA, Hughes DR. Associated groups for a semigroup. Bull Amer Math Soc 1952; 58: 624–625.
- [2] Steinfeld O. Uher die quasi ideals. Von halbgruppend Publ Math Debrecen 1956; 4: 262–275.
- [3] Grosek O, Satko L. A new notion in the theory of semigroup. Semigroup Forum 1980; 20: 233–240.
- [4] 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.
- [5] Wattanatripop K, Chinram R, Changphas T. Quasi-A-ideals and fuzzy A-ideals in semigroups. J Discrete Math Sci Cryptogr 2018; 21: 1131–1138.
- [6] Kaopusek N, Kaewnoi T, Chinram R. On almost interior ideals and weakly almost interior ideals of semigroups. J Discrete Math Sci Cryptogr 2020; 23: 773–778.
- [7] Iampan A, Chinram R, Petchkaew P. A note on almost subsemigroups of semigroups. Int J Math Comput Sci 2021; 16(4): 1623–1629.
- [8] 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.
- [9] Gaketem T. Almost bi-interior ideal in semigroups and their fuzzifications. Eur J Pure Appl Math 2022; 15(1): 281–289.
- [10] Gaketem T, Chinram R. Almost bi-quasi ideals and their fuzzifcations in semigroups. Ann Univ Craiova Math Comput Sci Ser 2023; 50(2): 42–352.
- [11] Wattanatripop K, Chinram R, Changphas T. Fuzzy almost bi-ideals in semigroups. Int J Math Comput Sci 2018; 13: 51–58.
- [12] Krailoet W, Simuen A, Chinram R, Petchkaew P. A note on fuzzy almost interior ideals in semigroups. Int J Math Comput Sci 2021; 16: 803–808.
- [13] Molodtsov D. Soft set theory-first results. Comput Math Appl 1999; 37(1): 19–31.
- [14] Maji PK, Biswas R, Roy AR. Soft set theory. Comput Math Appl 2003; 45(1): 555–562.
- [15] Pei D, Miao D. From soft sets to information systems. In: Proceedings of Granular Computing. IEEE 2005; 2: 617–621.
- [16] 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.
- [17] Sezgin A, Atagün AO. On operations of soft sets. Comput Math Appl 2011; 61(5):1457–1467.
- [18] Feng F, Jun YB, Zhao X. Soft semirings. Comput Math Appl 2008; 56(10): 2621–2628.
- [19] Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl 2011; 61: 2647–2654.
- [20] Sezgin A, Shahzad A, Mehmood A. New operation on soft sets: Extended difference of soft sets. J New Theory 2019; 27: 33–42.
- [21] Stojanovic NS. A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier 2021; 69(4): 779–791.
- [22] Sezgin A, Atagün AO. New soft set operation: Complementary soft binary piecewise plus operation. Matrix Science Mathematic 2023; 7(2): 125–142.
- [23] Sezgin A, Aybek FN. New soft set operation: Complementary soft binary piecewise gamma operation. Matrix Science Mathematic 2023; (7)1: 27–45.
- [24] Sezgin A, Aybek FN, Atagün AO. New soft set operation: Complementary soft binary piecewise intersection operation. BSJ Eng Sci 2023; 6(4): 330–346.
- [25] Sezgin A, Aybek FN, Güngör NB. New soft set operation: Complementary soft binary piecewise union operation. Acta Informatica Malaysia 2023; 7(1): 38–53.
- [26] Sezgin A, Demirci AM. New soft set operation: Complementary soft binary piecewise star operation. Ikonion Journal of Mathematics 2023; 5(2): 24–52.
- [27] Sezgin A, Yavuz E. New soft set operation: Complementary Soft Binary Piecewise Lambda Operation. Sinop University Journal of Natural Sciences 2023; 8(2): 101–133.
- [28] 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.
- [29] Sezgin A, Çağman N. New soft set operation: Complementary soft binary piecewise difference operation. Osmaniye Korkut Ata Üniv Fen Biliml Derg 2024; 7(1): 58–94.
- [30] Çağman N, Enginoğlu S. Soft set theory and uni-int decision making. Eur J Oper Res 2010; 20, 7(2): 848–855.
- [31] Çağman N, Çitak F, Aktaş H. Soft int-group and its applications to group theory. Neural Comput Appl 2012; 2: 151–158.
- [32] 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.
- [33] 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.
- [34] Sezgin A, Orbay M. Analysis of semigroups with soft intersection ideals. Acta Univ Sapientiae Math 2022; 14(1): 166–210.
- [35] Mahmood T, Rehman ZU, Sezgin A. Lattice ordered soft near rings. Korean J Math 2018; 26(3): 503–517.
- [36] Jana C, Pal M, Karaaslan F, Sezgin A. (α, β)-soft intersectional rings and ideals with their applications. New Math Nat Comput 2019; 15(2): 333–350.
- [37] Muştuoğlu E, Sezgin A, Türk ZK. Some characterizations on soft uni-groups and normal soft uni-groups. Int J Comput Appl 2016; 155(10): 1–8.
- [38] Sezer AS, Çağman N, Atagün AO. Uni-soft substructures of groups. Ann Fuzzy Math Inform 2015; 9(2): 235–246.
- [39] Sezer AS. Certain Characterizations of LA-semigroups by soft sets. J Intel Fuzzy Syst 2014; 27(2): 1035–1046.
- [40] Özlü Ş, Sezgin A. Soft covered ideals in semigroups. Acta Univ Sapientiae Math 2020; 12(2): 317–346.
- [41] Atagün AO, Sezgin A. Soft subnear-rings, soft ideals and soft n-subgroups of near-rings. Math Sci Letters 2018; 7(1): 37–42.
- [42] Sezgin A. A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat 2018; 32(8): 2995–3030.
- [43] Sezgin A, Çağman N, Atagün AO. A completely new view to soft intersection rings via soft uni-int product. Appl Soft Comput 2017; 54: 366–392.
- [44] Sezgin A, Atagün AO, Çağman N, Demir H. On near-rings with soft union ideals and applications. New Math Nat Comput 2022; 18(2): 495–511.
- [45] Rao MMK. Bi-interior ideals of semigroups. Discuss Math Gen Algebra Appl 2018; 38: 69–78.
- [46] Rao MMK. A study of a generalization of bi-ideal, quasi ideal and interior ideal of semigroup. Mathematica Moravica 2018; 22: 103–115.
- [47] Rao MMK. Left bi-quasi ideals of semigroups. Southeast Asian Bull Math 2020; 44: 369–376.
- [48] Rao MMK. Quasi-interior ideals and weak-interior ideals. Asia Pac Journal Mat 2020; 7(21): 1–20.
- [49] 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.
- [50] Sezgin A, Kocakaya FZ. Soft intersection quasi-interior ideals of semigroups. JuTISI 2024; (in press).
- [51] Sezgin A, İlgin A. Soft intersection almost subsemigroups of semigroups. Int J Math Phys 2024; 15(1): 13-20.
- [52] Sezgin A, İlgin A. Soft intersection almost ideals of semigroups. J Innovative Eng Nat Sci 2024; 4(2): 466-481.
- [53] Pant S, Dagtoros K, Kholil MI, Vivas A. Matrices: Peculiar determinant property. OPS Journal 2024, 1: 1–7.
- [54] Sezgin A, Çalışıcı H. A comprehensive study on soft binary piecewise difference operation. Eskişehir Teknik Univ Bilim Teknol Derg Teor Bilim 2024; 12(1): 32-54.
- [55] 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): 31-45.
- [56] Sezgin A, Sarıalioğlu M. New soft set operation: Complementary soft binary piecewise theta operation. Journal of Kadirli Faculty of Applied Sciences 2023; 4(2): 1-33.

Yıl 2024,
Cilt: 12 Sayı: 2, 81 - 99, 29.08.2024
### Öz

### Anahtar Kelimeler

### Kaynakça

Similar to how the quasi-interior ideal generalizes the ideal and interior ideal of a semigroup, the concept of soft intersection quasi-interior ideal generalizes the idea of soft intersection ideal and soft intersection interior ideal of a semigroup. In this study, we provide the notion of soft intersection almost quasi-interior ideal as well as the soft intersection weakly almost quasi-interior ideal in a semigroup. We show that any nonnull soft intersection quasi-interior ideal is a soft intersection almost quasi-interior ideal; and soft intersection almost quasi-interior ideal is a soft intersection weakly almost quasi-interior ideal, but the converses are not true. We further demonstrate that any idempotent soft intersection almost quasi-interior ideal is a soft intersection almost subsemigroup. With the established theorem that states that if a nonempty set A is almost quasi-interior ideal, then its soft characteristic function is a soft intersection almost quasi-interior ideal, and vice versa, we are also able to derive several intriguing relationships concerning minimality, primeness, semiprimeness, and strongly primeness between almost quasi-interior ideals, and soft intersection almost quasi-interior ideals.

Soft set semigroup (almost) quasi-interior ideals soft intersection (almost) quasi-interior ideals semigroup

- [1] Good RA, Hughes DR. Associated groups for a semigroup. Bull Amer Math Soc 1952; 58: 624–625.
- [2] Steinfeld O. Uher die quasi ideals. Von halbgruppend Publ Math Debrecen 1956; 4: 262–275.
- [3] Grosek O, Satko L. A new notion in the theory of semigroup. Semigroup Forum 1980; 20: 233–240.
- [4] 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.
- [5] Wattanatripop K, Chinram R, Changphas T. Quasi-A-ideals and fuzzy A-ideals in semigroups. J Discrete Math Sci Cryptogr 2018; 21: 1131–1138.
- [6] Kaopusek N, Kaewnoi T, Chinram R. On almost interior ideals and weakly almost interior ideals of semigroups. J Discrete Math Sci Cryptogr 2020; 23: 773–778.
- [7] Iampan A, Chinram R, Petchkaew P. A note on almost subsemigroups of semigroups. Int J Math Comput Sci 2021; 16(4): 1623–1629.
- [8] 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.
- [9] Gaketem T. Almost bi-interior ideal in semigroups and their fuzzifications. Eur J Pure Appl Math 2022; 15(1): 281–289.
- [10] Gaketem T, Chinram R. Almost bi-quasi ideals and their fuzzifcations in semigroups. Ann Univ Craiova Math Comput Sci Ser 2023; 50(2): 42–352.
- [11] Wattanatripop K, Chinram R, Changphas T. Fuzzy almost bi-ideals in semigroups. Int J Math Comput Sci 2018; 13: 51–58.
- [12] Krailoet W, Simuen A, Chinram R, Petchkaew P. A note on fuzzy almost interior ideals in semigroups. Int J Math Comput Sci 2021; 16: 803–808.
- [13] Molodtsov D. Soft set theory-first results. Comput Math Appl 1999; 37(1): 19–31.
- [14] Maji PK, Biswas R, Roy AR. Soft set theory. Comput Math Appl 2003; 45(1): 555–562.
- [15] Pei D, Miao D. From soft sets to information systems. In: Proceedings of Granular Computing. IEEE 2005; 2: 617–621.
- [16] 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.
- [17] Sezgin A, Atagün AO. On operations of soft sets. Comput Math Appl 2011; 61(5):1457–1467.
- [18] Feng F, Jun YB, Zhao X. Soft semirings. Comput Math Appl 2008; 56(10): 2621–2628.
- [19] Ali MI, Shabir M, Naz M. Algebraic structures of soft sets associated with new operations. Comput Math Appl 2011; 61: 2647–2654.
- [20] Sezgin A, Shahzad A, Mehmood A. New operation on soft sets: Extended difference of soft sets. J New Theory 2019; 27: 33–42.
- [21] Stojanovic NS. A new operation on soft sets: Extended symmetric difference of soft sets. Military Technical Courier 2021; 69(4): 779–791.
- [22] Sezgin A, Atagün AO. New soft set operation: Complementary soft binary piecewise plus operation. Matrix Science Mathematic 2023; 7(2): 125–142.
- [23] Sezgin A, Aybek FN. New soft set operation: Complementary soft binary piecewise gamma operation. Matrix Science Mathematic 2023; (7)1: 27–45.
- [24] Sezgin A, Aybek FN, Atagün AO. New soft set operation: Complementary soft binary piecewise intersection operation. BSJ Eng Sci 2023; 6(4): 330–346.
- [25] Sezgin A, Aybek FN, Güngör NB. New soft set operation: Complementary soft binary piecewise union operation. Acta Informatica Malaysia 2023; 7(1): 38–53.
- [26] Sezgin A, Demirci AM. New soft set operation: Complementary soft binary piecewise star operation. Ikonion Journal of Mathematics 2023; 5(2): 24–52.
- [27] Sezgin A, Yavuz E. New soft set operation: Complementary Soft Binary Piecewise Lambda Operation. Sinop University Journal of Natural Sciences 2023; 8(2): 101–133.
- [28] 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.
- [29] Sezgin A, Çağman N. New soft set operation: Complementary soft binary piecewise difference operation. Osmaniye Korkut Ata Üniv Fen Biliml Derg 2024; 7(1): 58–94.
- [30] Çağman N, Enginoğlu S. Soft set theory and uni-int decision making. Eur J Oper Res 2010; 20, 7(2): 848–855.
- [31] Çağman N, Çitak F, Aktaş H. Soft int-group and its applications to group theory. Neural Comput Appl 2012; 2: 151–158.
- [32] 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.
- [33] 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.
- [34] Sezgin A, Orbay M. Analysis of semigroups with soft intersection ideals. Acta Univ Sapientiae Math 2022; 14(1): 166–210.
- [35] Mahmood T, Rehman ZU, Sezgin A. Lattice ordered soft near rings. Korean J Math 2018; 26(3): 503–517.
- [36] Jana C, Pal M, Karaaslan F, Sezgin A. (α, β)-soft intersectional rings and ideals with their applications. New Math Nat Comput 2019; 15(2): 333–350.
- [37] Muştuoğlu E, Sezgin A, Türk ZK. Some characterizations on soft uni-groups and normal soft uni-groups. Int J Comput Appl 2016; 155(10): 1–8.
- [38] Sezer AS, Çağman N, Atagün AO. Uni-soft substructures of groups. Ann Fuzzy Math Inform 2015; 9(2): 235–246.
- [39] Sezer AS. Certain Characterizations of LA-semigroups by soft sets. J Intel Fuzzy Syst 2014; 27(2): 1035–1046.
- [40] Özlü Ş, Sezgin A. Soft covered ideals in semigroups. Acta Univ Sapientiae Math 2020; 12(2): 317–346.
- [41] Atagün AO, Sezgin A. Soft subnear-rings, soft ideals and soft n-subgroups of near-rings. Math Sci Letters 2018; 7(1): 37–42.
- [42] Sezgin A. A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat 2018; 32(8): 2995–3030.
- [43] Sezgin A, Çağman N, Atagün AO. A completely new view to soft intersection rings via soft uni-int product. Appl Soft Comput 2017; 54: 366–392.
- [44] Sezgin A, Atagün AO, Çağman N, Demir H. On near-rings with soft union ideals and applications. New Math Nat Comput 2022; 18(2): 495–511.
- [45] Rao MMK. Bi-interior ideals of semigroups. Discuss Math Gen Algebra Appl 2018; 38: 69–78.
- [46] Rao MMK. A study of a generalization of bi-ideal, quasi ideal and interior ideal of semigroup. Mathematica Moravica 2018; 22: 103–115.
- [47] Rao MMK. Left bi-quasi ideals of semigroups. Southeast Asian Bull Math 2020; 44: 369–376.
- [48] Rao MMK. Quasi-interior ideals and weak-interior ideals. Asia Pac Journal Mat 2020; 7(21): 1–20.
- [49] 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.
- [50] Sezgin A, Kocakaya FZ. Soft intersection quasi-interior ideals of semigroups. JuTISI 2024; (in press).
- [51] Sezgin A, İlgin A. Soft intersection almost subsemigroups of semigroups. Int J Math Phys 2024; 15(1): 13-20.
- [52] Sezgin A, İlgin A. Soft intersection almost ideals of semigroups. J Innovative Eng Nat Sci 2024; 4(2): 466-481.
- [53] Pant S, Dagtoros K, Kholil MI, Vivas A. Matrices: Peculiar determinant property. OPS Journal 2024, 1: 1–7.
- [54] Sezgin A, Çalışıcı H. A comprehensive study on soft binary piecewise difference operation. Eskişehir Teknik Univ Bilim Teknol Derg Teor Bilim 2024; 12(1): 32-54.
- [55] 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): 31-45.
- [56] Sezgin A, Sarıalioğlu M. New soft set operation: Complementary soft binary piecewise theta operation. Journal of Kadirli Faculty of Applied Sciences 2023; 4(2): 1-33.

Toplam 56 adet kaynakça vardır.

Birincil Dil | İngilizce |
---|---|

Konular | Cebir ve Sayı Teorisi |

Bölüm | Makaleler |

Yazarlar | |

Yayımlanma Tarihi | 29 Ağustos 2024 |

Gönderilme Tarihi | 25 Nisan 2024 |

Kabul Tarihi | 13 Haziran 2024 |

Yayımlandığı Sayı | Yıl 2024 Cilt: 12 Sayı: 2 |