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YARIGRUPLARIN ESNEK KESİŞİMSEL HEMEN HEMEN QUASİ-İÇ İDEALLERİ

Year 2024, Volume: 12 Issue: 2, 81 - 99, 29.08.2024

Aslıhan Sezgin , Fatıma Zehra Kocakaya , Aleyna İlgin

https://doi.org/10.20290/estubtdb.1473840

Abstract

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.

Keywords

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

References

  • [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.

SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS

Year 2024, Volume: 12 Issue: 2, 81 - 99, 29.08.2024

Aslıhan Sezgin , Fatıma Zehra Kocakaya , Aleyna İlgin

https://doi.org/10.20290/estubtdb.1473840

Abstract

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.

Keywords

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

References

  • [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.
There are 56 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Aslıhan Sezgin 0000-0002-1519-7294

Fatıma Zehra Kocakaya 0009-0002-0409-5556

Aleyna İlgin 0009-0001-5641-5462

Publication Date August 29, 2024
Submission Date April 25, 2024
Acceptance Date June 13, 2024
Published in Issue Year 2024 Volume: 12 Issue: 2

Cite

APA Sezgin, A., Kocakaya, F. Z., & İlgin, A. (2024). SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS. Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler, 12(2), 81-99. https://doi.org/10.20290/estubtdb.1473840
AMA Sezgin A, Kocakaya FZ, İlgin A. SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler. August 2024;12(2):81-99. doi:10.20290/estubtdb.1473840
Chicago Sezgin, Aslıhan, Fatıma Zehra Kocakaya, and Aleyna İlgin. “SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS”. Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler 12, no. 2 (August 2024): 81-99. https://doi.org/10.20290/estubtdb.1473840.
EndNote Sezgin A, Kocakaya FZ, İlgin A (August 1, 2024) SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 12 2 81–99.
IEEE A. Sezgin, F. Z. Kocakaya, and A. İlgin, “SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS”, Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler, vol. 12, no. 2, pp. 81–99, 2024, doi: 10.20290/estubtdb.1473840.
ISNAD Sezgin, Aslıhan et al. “SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS”. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler 12/2 (August 2024), 81-99. https://doi.org/10.20290/estubtdb.1473840.
JAMA Sezgin A, Kocakaya FZ, İlgin A. SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler. 2024;12:81–99.
MLA Sezgin, Aslıhan et al. “SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS”. Eskişehir Teknik Üniversitesi Bilim Ve Teknoloji Dergisi B - Teorik Bilimler, vol. 12, no. 2, 2024, pp. 81-99, doi:10.20290/estubtdb.1473840.
Vancouver Sezgin A, Kocakaya FZ, İlgin A. SOFT INTERSECTION ALMOST QUASI-INTERIOR IDEALS OF SEMIGROUPS. Eskişehir Teknik Üniversitesi Bilim ve Teknoloji Dergisi B - Teorik Bilimler. 2024;12(2):81-99.