Soft union bi-quasi-interior ideals of semigroups
Abstract
The concept of the soft union (S-uni) bi-quasi-interior ₿ꝖĪ) ideal of semigroups is proposed in this study, along with its equivalent definition. We derive the relationships between S-uni ideals and S-uni ₿ꝖĪ ideal. The S-uni ₿ꝖĪ ideal is shown to be S-uni bi-ideal, left ideal, right ideal, interior ideal, quasi-ideal, bi-interior ideal, left/right bi-quasi ideal, and left/right quasi-interior ideal. It is shown that certain additional requirements, such as regularity or right/left simplicity, are necessary for the converses, and counterexamples are given to demonstrate that the converses are not true. Additionally, it is demonstrated that the soft anti characteristic function of a subsemigroup of a semigroup is an S-uni ₿ꝖĪ ideal if the subsemigroup itself is a ₿ꝖĪ ideal, and vice versa. Consequently, a significant connection between soft set theory and classical semigroup theory is established. Additionally, it is demonstrated that while the finite soft OR-products and union of S-uni ₿ꝖĪ ideals are also S-uni ₿ꝖĪ ideals, the intersection and finite soft AND-products are not. A broad conceptual characterization and analysis of S-uni ₿ꝖĪ ideals are presented in this paper.
Keywords
Full Text:
PDFReferences
Good, R. A., & Hughes, D. R. (1952). Associated groups for a semigroup. Bulletin of the American Mathematical Society, 58(6), 624–625.
Steinfeld, O. (1956). Uher die quasi ideals. Von halbgruppend Publication Mathematical Debrecen, 4, 262–275.
Lajos S. (1976). (m;k;n)-ideals in semigroups. Notes on semigroups II, Karl Marx University of Economics Department of Mathematics Budapest, (1), 12–19.
Szasz G. (1977). Interior ideals in semigroups. Notes on semigroups IV. Karl Marx University of Economics Department of Mathematics Budapest, (5), 1–7.
Szasz G. (1981). Remark on interior ideals of semigroups. Studia Scientiarum Mathematicarum Hungarica, (16), 61–63.
Rao, M. M. K. (2018). Bi-interior ideals of semigroups. Discussiones Mathematicae-general Algebra and Applications, 38(1), 69–78. DOI:10.7151/dmgaa.1283
Rao, M. M. K. (2018). A study of a generalization of bi-ideal, quasi-ideal and interior ideal of semigroup. Mathematica Moravica, 22(2), 103–115. DOI:10.5937/MatMor1802103M
Rao, M. M. K. (2020). Left bi-quasi ideals of semigroups. Southeast Asian Bulletion of Mathematics, 44(3), 369–376. DOI:10.7251/BIMVI1801045R
Rao, M. M. K. (2020). Quasi-interior ideals and weak-interior ideals. Asia Pacific Journal Mathematical, 7, 7–21. DOI: 10.28924/APJM/7-21
Rao, M. M. K. (2024). Tri-ideals of semigroups. Bulletin of International Mathematical Virtual Institute. 2024. 14(2): 213-223. DOI: 10.7251/BIMVI2402213M
Rao, M. M. K., Kona, R. K., Rafi, N., Bolineni, V. (2024). Tri-quasi ideals and fuzzy tri-quasi ideals of semigroups. Annals of Communications in Mathematics. 7(3): 281-295. DOI: 10.62072/acm.2024.070307
Baupradist, S., Chemat, B., Palanivel, K., & Chinram, R. (2021). Essential ideals and essential fuzzy ideals in semigroups. Journal of Discrete Mathematical Sciences and Cryptography, 24(1), 223–233. https://doi.org/10.1080/09720529.2020.1816643
Grošek, O., & Satko, L. (1980). A new notion in the theory of semigroup. Semigroup Forum, 20(1), 233–240, https://doi.org/10.1007/BF02572683
Bogdanovic, S. (1981). Semigroups in which some bi-ideal is a group. Zbornik radova PMF Novi Sad, 11, 261–266.
Wattanatripop, K., Chinram, R., & Changphas, T. (2018). Quasi-A-ideals and fuzzy A-ideals in semigroups. Journal of Discrete Mathematical Sciences and Cryptography, 21(5), 1131–1138. DOI:10.1080/09720529.2018.1468608
Kaopusek, N., Kaewnoi, T., & Chinram, R. (2020). On almost interior ideals and weakly almost interior ideals of semigroups. Journal of Discrete Mathematical Sciences and Cryptography, 23(3), 773–778. https://doi.org/10.1080/09720529.2019.1696917
Iampan, A., Chinram, R., & Petchkaew, P. (2021). A note on almost subsemigroups of semigroups. International Journal Mathematical Computer Science, 16, 1623–1629, https://future-in-tech.net/16.4/R-Petchkaew.pdf
Chinram, R., & Nakkhasen, W. (2021). Almost bi-quasi-interior ideals and fuzzy almost bi-quasi-interior ideals of semigroups. Journal Mathematical Computer Science, 26, 128–136. DOI: 10.22436/jmcs.026.02.03
Gaketem, T. (2022). Almost bi-interior ideal in semigroups and their fuzzifications. European Journal of Pure and Applied Mathematics, 15(1), 281–289. https://doi.org/10.29020/nybg.ejpam.v15i1.4279
Gaketem, T., & Chinram, R. (2023). Almost bi-quasi ideals and their fuzzifications in semigroups. Annals of the University of Craiova-Mathematicsa and Computer Science Series, 50(2), 342–352. https://doi.org/10.52846/ami.v50i2.1708
Wattanatripop, K., Chinram, R., & Changphas, T. (2018). Fuzzy almost bi-ideals in semigroups. International Journal of Mathematics and Computer Science, 13(1), 51–58. https://future-in-tech.net/13.1/R-2Chinram.pdf
Krailoet, W., Simuen, A., Chinram, R., & Petchkaew, P. (2021). A note on fuzzy almost interior ideals in semigroups. International Journal of Mathematics and Computer Science, 16(2), 803–808. https://future-in-tech.net/16.2/R-Pattarawan-Petchkaew.pdf
Molodtsov, D. (1999). Soft set theory—first results. Computers & Mathematics with Applications, 37(4–5), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
Maji, P. K., Biswas, R., & Roy, A. R. (2003). Soft set theory. Computers & Mathematics with Applications, 45(4–5), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
Pei, D., & Miao, D. (2005). From soft sets to information systems. 2005 IEEE International Conference on Granular Computing (Vol. 2, pp. 617–621). IEEE.
Ali, M. I., Feng, F., Liu, X., Min, W. K., & Shabir, M. (2009). On some new operations in soft set theory. Computers & Mathematics with Applications, 57(9), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009
Sezgin, A., & Atagün, A. O. (2011). On operations of soft sets. Computers & Mathematics with Applications, 61(5), 1457–1467. https://doi.org/10.1016/j.camwa.2011.01.018
Sezgin, A., & Sarıalioğlu, M. (2024). A new Soft Set Operation: Complementary Soft Binary Piecewise Theta (
Sezgin, A. and Dagtoros, K. (2023). Complementary soft binary piecewise symmetric difference operation: A novel soft set operation. Scientific Journal of Mehmet Akif Ersoy University, 6(2): 31-45.https://dergipark.org.tr/en/pub/sjmakeu/issue/82332/1365021
Sezgin, A. and Çalışıcı, H. (2024). A comprehensive study on soft binary piecewise difference operation. Eskişehir Technical University Journal of Science and Technology B- Theoretical Sciences. 12(1): 32-54. https://doi.org/10.20290/estubtdb.1356881
Sezgin, A., Çağman, N., Atagün, A. O. and Aybek, F. N. (2023). Complemental binary operations of sets and their application to group theory. Matrix Science Mathematic, 7(2): 114-121. DOI: http://doi.org/10.26480/msmk.02.2023.114.121
Sezgin, A. and Yavuz, E. (2024). Soft binary piecewise plus operation: A new type of operation for soft sets. Uncertainty Discourse and Applications, 1(1): 79-100. https://uda.reapress.com/journal/article/view/26
Sezgin, A. and Atagün A. O. and Çağman N. A complete study on and-product of soft sets, Sigma Journal of Engineering and Natural Sciences, in press
Sezgin, A., Aybek, F. N., & Atagün, A. O. (2023). A new soft set operation: complementary soft binary piecewise intersection operation. Black Sea Journal of Engineering and Science, 6(4), 330–346. https://doi.org/10.34248/bsengineering.1319873
Sezgin, A., Aybek, F. N., & Güngör, N. B. (2023). A new soft set operation: complementary soft binary piecewise union operation. Acta Informatica Malaysia, 7(1), 38–53. http://doi.org/10.26480/aim.01.2023.38.53
Sezgin, A., & Demirci, A. M. (2023). A new soft set operation: complementary soft binary piecewise star operation. Ikonion Journal of Mathematics, 5(2), 24–52. https://doi.org/10.54286/ikjm.1304566
Sezgin, A., & Yavuz, E. (2023). A new soft set operation: complementary soft binary piecewise lambda operation. Sinop University Journal of Natural Sciences, 8(5), 101–133. https://doi.org/10.33484/sinopfbd.1320420
Sezgin, A., & Yavuz, E. (2023). A new soft set operation: soft binary piecewise symmetric difference operation. Necmettin Erbakan University Journal of Science and Engineering, 5(2), 189–208. https://doi.org/10.47112/neufmbd.2023.18
Sezgin, A., & Çağman, N. (2024). A new soft set operation: complementary soft binary piecewise difference operation. Osmaniye Korkut Ata University Journal of The Institute of Science and Technology, 7(1), 58–94. https://doi.org/10.47495/okufbed.1308379
Çağman, N., & Enginoğlu, S. (2010). Soft set theory and uni-int decision making. European Journal of Operational Research, 207(2), 848–855. https://doi.org/10.1016/j.ejor.2010.05.004
Çağman, N., Çıtak, F., & Aktaş, H. (2012). Soft int-group and its applications to group theory. Neural Computing and Applications, 21, 151–158. https://doi.org/10.1007/s00521-011-0752-x
Sezgin, A. (2016). A new approach to semigroup theory I: Soft union semigroups, ideals and bi-ideals, Algebra Letters, Article ID 3, 46 pages. https://scik.org/index.php/abl/article/view/2989
Sezgin, A., Çağman, N. and Atagün, A.O. A New Approach to Semigroup Theory II; Soft Union Interior Ideals, Quasi-ideals and Generalized Bi-ideals, Thai Journal of Mathematics, in press.
Sezer, A. S., Çağman, N., Atagün, A. O. (2015). A Novel Characterization for Certain Semigroups by Soft Union Ideals, Information Sciences Letters, 4(1), 13-18.
Sezgin, A. and Orbay, K. (2023). Completely weakly, quasi-regular semigroups characterized by soft union Quasi ideals, (generalized) bi-ideals and semiprime ideals, Sigma: Journal of Engineering & Natural Sciences, 41(4), 868−874. DOI: 10.14744/sigma.2023.00093
Sezgin, A., & İlgin, A. (2024). Soft intersection almost ideals of semigroups. Journal of Innovative Engineering and Natural Science, 4(2), 466-481. https://doi.org/10.61112/jiens.1464344
Sezgin, A., & İlgin, A. (2024). Soft intersection almost bi-interior ideals of semigroups. Journal of Natural and Applied Sciences Pakistan, 6(1), 1619-1638. https://jnasp.kinnaird.edu.pk/wp-content/uploads/2024/08/3.Aslihan-SezginJNASP-2024-0290.pdf
Sezgin, A., & İlgin, A. (2024). Soft intersection almost bi-quasi ideals of semigroups. Soft Computing Fusion with Applications, 1(1), 27-42. https://doi.org/10.22105/dm20a128
Sezgin, A., & İlgin, A. (2024). Soft intersection almost weak-interior ideals of semigroups: a theoretical study. JNSM Journal of Natural Sciences and Mathematics of UT, 9(17-18), 372–385. https://doi.org/10.62792/ut.jnsm.v9.i17-18.p2834
Sezgin, A., & Onur, B. (2024). Soft intersection Almost Bi-ideals of Semigroups. Systemic Analytics, 2(1), 95-105. https://doi.org/10.31181/sa21202415
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
Sezgin, A., & İlgin, A. (2024). Soft intersection almost subsemigroups of semigroups. International Journal of Mathematics and Physics, 15(1), 13-20. https://doi.org/10.26577/ijmph.2024v15i1a2
Sezgin, A., Baş, Z. H. & İlgin, A. (2025). Soft intersection almost bi-quasi-interior ideals of semigroups. Journal of Fuzzy Extension and Application, 6(1), 43-58. https://doi.org/10.22105/jfea.2024.452790.1445
Sezgin, A., Onur, B., & İlgin, A. (2024). Soft intersection almost tri-ideals of semigroups. SciNexuses, 1, 126-138. https://doi.org/10.61356/j.scin.2024.1414
Sezgin, A., Ilgin, A., & Atagün, A. O. (2024). Soft intersection almost tri-bi-ideals of semigroups. Science & Technology Asia, 29(4), 1-13. Retrieved from https://ph02.tci-thaijo.org/index.php/SciTechAsia/article/view/253582
Sezgin, A., & Kocakaya, F. Z. (2025). Soft intersection almost quasi-ideals of semigroups. songklanakarin journal of science and technology, in press.
Sezgin, A., & Baş, Z. H. (2024). Soft-int almost interior ıdeals for semigroups. Information Sciences with Applications, 4, 25-36. https://doi.org/10.61356/j.iswa.2024.4374
Atagün, A. O, Kamacı, H, Taştekin, İ, & Sezgin, A. (2019). P-properties in near-rings. Journal of Mathematical and Fundamental Sciences, 51(2): 152-167. https://dx.doi.org/10.5614/j.math.fund.sci.2019.51.2.5
Jana, C., Pal, M., Karaaslan, F., & Sezgi̇n, A. (2019). (α, β)-Soft intersectional rings and ideals with their applications. New Mathematics and Natural Computation, 15(02), 333–350. https://doi.org/10.1142/S1793005719500182
Manikantan, T., Ramasany, P., & Sezgin, A. (2023). Soft quasi-ideals of soft near-rings. Sigma Journal of Engineering and Natural Science, 41(3), 565-574. DOI: 10.14744/sigma.2023.00062
Khan, A., Izhar, M, Sezgin, A. (2017). Characterizations of Abel Grassmann’s groupoids by the properties of double-framed soft ideals. International Journal of Analysis and Applications, 15(1), 62-74.
Sezer, A. S., Çağman, N., & Atagün, A. O. (2015). Uni-soft substructures of groups. Annals of Fuzzy Mathematics And Informatics, 9(2), 235–246. http://www.afmi.or.kr/papers/2015/Vol-09_No-02/PDF/AFMI-9-2(235-246)-H-140701R2.pdf
Gulistan, M., Feng, F., Khan, M., Sezgin, & A. (2018). Characterizations of right weakly regular semigroups in terms of generalized cubic soft sets. Mathematics, 6, 293. https://doi.org/10.3390/math6120293
Atagün, A. O., & Sezgin, A. (2018). Soft subnear-rings, soft ideals and soft N-subgroups of near-rings. Mathematical Science Letter, 7, 37–42. http://dx.doi.org/10.18576/msl/070106
Sezgin, A. (2018). A new view on AG-groupoid theory via soft sets for uncertainty modeling. Filomat, 32(8), 2995–3030. https://doi.org/10.2298/FIL1808995S
Sezgin, A., Çağman, N., & Atagün, A. O. (2017). A completely new view to soft-int rings via soft uni-int product. Applied Soft Computing, 54, 366–392. https://doi.org/10.1016/j.asoc.2016.10.004
Sezgin, A., Atagün, A. O., Çağman, N., & Demir, H. (2022). On near-rings with soft union ideals and applications. New Mathematics And Natural Computation, 18(2), 495–511. DOI: 10.1142/S1793005722500247
Atagün, A. O., & Sezgin, A.(2017). Int-soft substructures of groups and semirings with applications. Applied Mathematics & Information Sciences, 11(1), 105-113. doi:10.18576/amis/110113
Sezer, A.S., Atagün, A.O., & Çağman, N. (2014). N-group SI-action and its applications to N-group theory. Fasciculi Mathematici, 52, 139-153.
Sezgin, A.S., İlgin, A., Kocakaya, F.Z., Baş, Z.H., Onur, B., & Çıtak, F. (2024). A remarkable contribution to soft int-group theory via a comprehensive view on soft cosets, Journal of Science and Arts, 24(4), 935–934. https://www.josa.ro/docs/josa_2024_4/a_13_Sezgin_905-934_30p.pdf
Rao, M. M. K. A Study of bi-quasi-interior ideal as a new generalization of ideal of generalization of semiring. Journal of International Mathematical Virtual Institute. 2019. 9: 19-35.
Rao, M. M. K. Bi-quasi-interior Ideals. Annals of Communications in Mathematics. 2024. 7(3): 296-309.
Rao, M. M. K. Bi-quasi-interior Ideals of Semiring. Journal of International Mathematical Virtual Institute. 2024. 14(1): 17-31.
Clifford, A. H. and Preston, G. B. The Algebraic Theory of Semigroups Vol. I (Second Edition). American Mathematical Society. 1964.
İlgin. A. & Sezgin, A. (2024). Soft union weak-interior ideals of semigroups. International Journal of Open Problems in Computer Science and Mathematics, 18 (2),1-23
Kocakaya, F.Z. & Sezgin, A., Soft union quasi-interior ideals of semigroups International Journal of Mathematics and Physics, in press..
İlgin. A. & Sezgin, A. (2024). Soft union bi-interior ideals of semigroups. International Journal of Advances in Engineering and Pure Sciences, in press.
Onur, B. & Sezgin, A. (2024). Soft union bi-quasi ideals of semigroups. Turkish Journal of Nature and Science, in press.
DOI: https://doi.org/10.24198/jmi.v21.n1.62779.55-74
Refbacks
- There are currently no refbacks.
Copyright (c) 2025 Jurnal Matematika Integratif
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Published By:
Department of Matematics, FMIPA, Universitas Padjadjaran, Jl. Raya Bandung-Sumedang KM. 21 Jatinangor
Indexed by:
Visitor Number : View My Stats
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.