Research Article
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Set-generated soft subrings of rings

Year 2022, Volume: 71 Issue: 4, 993 - 1006, 30.12.2022

Akın Osman Atagün , Hüseyin Kamacı

https://doi.org/10.31801/cfsuasmas.1013172

Abstract

This paper focuses on the set-oriented operations and set-oriented algebraic structures of soft sets. Relatedly, in this paper, firstly some essential properties of $\alpha$-intersection of soft set are investigated, where $\alpha$ is a non-empty subset of the universal set. Later, by using $\alpha$-intersection of soft set, the notion of set-generated soft subring of a ring is introduced. The generators of soft intersections and products of soft subrings are given. Some related properties about generators of soft subrings are investigated and illustrated by several examples.

Keywords

Soft sets soft subrings of rings generated soft subrings of rings

References

  • Zadeh, L. A., Fuzzy sets, Inf. Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
  • Zadeh, L. A., Toward a generalized theory of uncertainty (GTU)-an outline, Inf. Sci., 172 (2005), 1–40. https://doi.org/10.1016/j.ins.2005.01.017
  • Gorzalzany, M. B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets Syst., 21 (1987), 1–17. https://doi.org/10.1016/0165-0114(87)90148-5
  • Gau, W. L., Buehrer, D. J., Vague sets, IEEE Trans. Syst. Man Cybern., 23 (1993), 610–614. doi: 10.1109/21.229476
  • Pawlak, Z., Rough sets, Int. J. Comput. Inf. Sci., 11 (1982), 341–356. http://dx.doi.org/10.1007/BF01001956
  • Pawlak, Z., Skowron, A., Rudiments of rough sets, Inf. Sci., 177 (2007), 3–27. doi:10.1016/j.ins.2006.06.003
  • Molodtsov, D., Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
  • Maji, P. K., Biswas, R., Roy, A. R., Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
  • Maji, P. K., Roy, A. R., Biswas, R., An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X
  • Ali, M. I., Feng, F., Liu, X., Min, W. K., Shabir, M., On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009
  • Çağman, N., Enginoğlu, S., Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207 (2010), 848–855. https://doi.org/10.1016/j.ejor.2010.05.004
  • Kamacı, H., Similarity measure for soft matrices and its applications, J. Intell. Fuzzy Syst., 36 (2019), 3061–3072. doi: 10.3233/JIFS-18339
  • Kamacı, H., Atagün, A. O., Aygün, E., Difference operations of soft matrices with applications in decision making, Punjab Univ. J. Math., 51 (2019), 1–21.
  • Sezgin, A., Atagün, A. O., On operations of soft sets, Comput. Math. Appl., 61 (2011), 1457–1467.https://doi.org/10.1016/j.camwa.2011.01.018
  • Aygün, E., Kamacı, H., Some generalized operations in soft set theory and their role in similarity and decision making, J. Intell. Fuzzy Syst., 36 (2019), 6537–6547. doi: 10.3233/JIFS-182924
  • Aygün, E., Kamacı, H., Some new algebraic structures of soft sets, Soft Comput., 25(13) (2021), 8609–8626. https://doi.org/10.1007/s00500-021-05744-y
  • Çağman, N., Enginoğlu, S., Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010), 3308–3314. https://doi.org/10.1016/j.camwa.2010.03.015
  • Atagün, A. O., Kamacı, H., Oktay, O., Reduced soft matrices and generalized products with applications in decision making, Neural Comput. Appl., 29 (2018), 445–456. https://doi.org/10.1007/s00521-016-2542-y
  • Kamacı, H., Atagün, A. O., Sönmezoğlu, A., Row-products of soft matrices with applications in multiple-disjoint decision making, Appl. Soft Comput., 62 (2018), 892–914. https://doi.org/10.1016/j.asoc.2017.09.024
  • Kamacı, H., Atagün, A. O., Toktaş, E., Bijective soft matrix theory and multi-bijective linguistic soft decision system, Filomat, 32 (2018), 3799–3814. https://doi.org/10.2298/FIL1811799K
  • Petchimuthu, S., Garg, H., Kamacı, H., Atagün, A. O., The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM, Comput. Appl. Math., 39 (2020), Article Number 68. https://doi.org/10.1007/s40314-020-1083-2
  • Kamacı, H., Saltık, K., Akız, H. F., Atagün, A. O., Cardinality inverse soft matrix theory and its applications in multicriteria group decision making, J. Intell. Fuzzy Syst., 34 (2018), 2031–2049. doi: 10.3233/JIFS-17876
  • Petchimuthu, S., Kamacı, H., The row-products of inverse soft matrices in multicriteria decision making, J. Intell. Fuzzy Syst., 36 (2019), 6425–6441. doi: 10.3233/JIFS-182709
  • Aktaş, H., Çağman, N., Soft sets and soft groups, Inf. Sci., 177 (2007), 2726–2735. https://doi.org/10.1016/j.ins.2006.12.008
  • Ulucay, V., Oztekin, O., Sahin, M., Olgun, N., Kargin, A., Soft representation of soft groups, New Trend Math. Sci., 4(2) (2016), 23. http://dx.doi.org/10.20852/ntmsci.2016217001
  • Feng, F., Jun, Y. B., Zhao, X., Soft semirings, Comput. Math. Appl., 56 (2008), 2621–2628. https://doi.org/10.1016/j.camwa.2008.05.011
  • Acar, U., Koyuncu, F., Tanay, B., Soft sets and soft rings, Comput. Math. Appl., 59 (2010), 3458-3463. https://doi.org/10.1016/j.camwa.2010.03.034
  • Uluçay, V., Şahin, M., Olgun, N., Soft normed rings, SpringerPlus, 5(1) (2016), 1–6. doi: 10.1186/s40064-016-3636-9
  • Atagün, A. O., Sezer, A. S., Soft substructures of rings fields and modules, Comput. Math. Appl., 61 (2011), 592–601. https://doi.org/10.1016/j.camwa.2010.12.005
  • Sezgin, A., Atagün, A. O., Aygün, E., A note on soft near-rings and idealistic soft near-rings, Filomat, 25 (2011), 53–68. doi: 10.2298/FIL1101053S
  • Ostadhadi-Dehkordi, S., Shum, K. P., Regular and strongly regular relations on soft hyperrings, Soft Comput., 23 (2019), 3253–3260. https://doi.org/10.1007/s00500-018-03711-8
  • Tahat, M. K., Sidky, F., Abo-Elhamayel, M., Soft topological soft groups and soft rings, Soft Comput., 22 (2018), 7143–7156. https://doi.org/10.1007/s00500-018-3026-z
  • Karaaslan, F., Some properties of AG*-groupoids and AG-bands under SI-product operation, J. Intell. Fuzzy Syst., 36 (2019), 231–239. doi: 10.3233/JIFS-181208
  • Yousafzaia, F., Khalaf, M. M., Alia, A., Arsham B., Saeidc, D., Non-associative ordered semigroups based on soft sets, Commun. Algebra, 47 (2019), 312–327. https://doi.org/10.1080/00927872.2018.1476524
  • Zhan, J., Dudek, W. A., Neggers, J., A new soft union set: characterizations of hemirings, Int. J. Mach. Learn. Cybern., 8 (2017), 525–535. https://doi.org/10.1007/s13042-015-0343-8
  • Atagün, A. O., Sezgin, A., Soft subnear-rings, soft ideals and soft N-subgroups of near-rings, Math. Sci. Lett., 7 (2018), 37–42. http://dx.doi.org/10.18576/msl/070106
  • Riaz, M., Naeem, K., Aslam, M., Afzal, D., Almahdi, F. A. A., Jamal, S. S., Multi-criteria group decision making with Pythagorean fuzzy soft topology, J. Intell. Fuzzy Syst., 39 (2020), 6703–6720. doi: 10.3233/JIFS-190854
  • Riaz, M., Naim, Ç., Zareef, I., Aslam, M., N-soft topology and its applications to multi-criteria group decision making, J. Intell. Fuzzy Syst., 36 (2019), 6521–6536. doi: 10.3233/JIFS-182919
  • Riaz, M., Tehreim, S. T., On bipolar fuzzy soft topology with decision-making, Soft Comput., 24 (2020), 18259–18272. https://doi.org/10.1007/s00500-020-05342-4
  • Sezer, A. S., Çağman, N., Atagün, A. O., Ali, M. I., Türkmen, E., Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I, Filomat, 29 (2015), 917–946. doi: 10.2298/FIL1505917S
  • Sezgin, A., Çağman, N., Çıtak, F., α-inclusions applied to group theory via soft set and logic, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68 (2019), 334–352. doi: 10.31801/cfsuasmas.420457
  • Feng, F., Li, C. X., Davvaz, B., Ali, M. I., Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14 (2010), 899–911. https://doi.org/10.1007/s00500-009-0465-6
  • Feng, F., Liu, X. Y., Leoreanu-Fotea, V., Jun, Y. B., Soft sets and soft rough sets, Inf. Sci., 181 (2011), 1125–1137. https://doi.org/10.1016/j.ins.2010.11.004
  • Atagün, A. O., Kamacı, H., Decompositions of soft sets and soft matrices with applications in group decision making, Scientia Iranica, in press (2021). doi:10.24200/SCI.2021.58119.5575.

Year 2022, Volume: 71 Issue: 4, 993 - 1006, 30.12.2022

Akın Osman Atagün , Hüseyin Kamacı

https://doi.org/10.31801/cfsuasmas.1013172

Abstract

References

  • Zadeh, L. A., Fuzzy sets, Inf. Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X
  • Zadeh, L. A., Toward a generalized theory of uncertainty (GTU)-an outline, Inf. Sci., 172 (2005), 1–40. https://doi.org/10.1016/j.ins.2005.01.017
  • Gorzalzany, M. B., A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets Syst., 21 (1987), 1–17. https://doi.org/10.1016/0165-0114(87)90148-5
  • Gau, W. L., Buehrer, D. J., Vague sets, IEEE Trans. Syst. Man Cybern., 23 (1993), 610–614. doi: 10.1109/21.229476
  • Pawlak, Z., Rough sets, Int. J. Comput. Inf. Sci., 11 (1982), 341–356. http://dx.doi.org/10.1007/BF01001956
  • Pawlak, Z., Skowron, A., Rudiments of rough sets, Inf. Sci., 177 (2007), 3–27. doi:10.1016/j.ins.2006.06.003
  • Molodtsov, D., Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5
  • Maji, P. K., Biswas, R., Roy, A. R., Soft set theory, Comput. Math. Appl., 45 (2003), 555–562. https://doi.org/10.1016/S0898-1221(03)00016-6
  • Maji, P. K., Roy, A. R., Biswas, R., An application of soft sets in a decision making problem, Comput. Math. Appl., 44 (2002), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X
  • Ali, M. I., Feng, F., Liu, X., Min, W. K., Shabir, M., On some new operations in soft set theory, Comput. Math. Appl., 57 (2009), 1547–1553. https://doi.org/10.1016/j.camwa.2008.11.009
  • Çağman, N., Enginoğlu, S., Soft set theory and uni-int decision making, Eur. J. Oper. Res., 207 (2010), 848–855. https://doi.org/10.1016/j.ejor.2010.05.004
  • Kamacı, H., Similarity measure for soft matrices and its applications, J. Intell. Fuzzy Syst., 36 (2019), 3061–3072. doi: 10.3233/JIFS-18339
  • Kamacı, H., Atagün, A. O., Aygün, E., Difference operations of soft matrices with applications in decision making, Punjab Univ. J. Math., 51 (2019), 1–21.
  • Sezgin, A., Atagün, A. O., On operations of soft sets, Comput. Math. Appl., 61 (2011), 1457–1467.https://doi.org/10.1016/j.camwa.2011.01.018
  • Aygün, E., Kamacı, H., Some generalized operations in soft set theory and their role in similarity and decision making, J. Intell. Fuzzy Syst., 36 (2019), 6537–6547. doi: 10.3233/JIFS-182924
  • Aygün, E., Kamacı, H., Some new algebraic structures of soft sets, Soft Comput., 25(13) (2021), 8609–8626. https://doi.org/10.1007/s00500-021-05744-y
  • Çağman, N., Enginoğlu, S., Soft matrix theory and its decision making, Comput. Math. Appl., 59 (2010), 3308–3314. https://doi.org/10.1016/j.camwa.2010.03.015
  • Atagün, A. O., Kamacı, H., Oktay, O., Reduced soft matrices and generalized products with applications in decision making, Neural Comput. Appl., 29 (2018), 445–456. https://doi.org/10.1007/s00521-016-2542-y
  • Kamacı, H., Atagün, A. O., Sönmezoğlu, A., Row-products of soft matrices with applications in multiple-disjoint decision making, Appl. Soft Comput., 62 (2018), 892–914. https://doi.org/10.1016/j.asoc.2017.09.024
  • Kamacı, H., Atagün, A. O., Toktaş, E., Bijective soft matrix theory and multi-bijective linguistic soft decision system, Filomat, 32 (2018), 3799–3814. https://doi.org/10.2298/FIL1811799K
  • Petchimuthu, S., Garg, H., Kamacı, H., Atagün, A. O., The mean operators and generalized products of fuzzy soft matrices and their applications in MCGDM, Comput. Appl. Math., 39 (2020), Article Number 68. https://doi.org/10.1007/s40314-020-1083-2
  • Kamacı, H., Saltık, K., Akız, H. F., Atagün, A. O., Cardinality inverse soft matrix theory and its applications in multicriteria group decision making, J. Intell. Fuzzy Syst., 34 (2018), 2031–2049. doi: 10.3233/JIFS-17876
  • Petchimuthu, S., Kamacı, H., The row-products of inverse soft matrices in multicriteria decision making, J. Intell. Fuzzy Syst., 36 (2019), 6425–6441. doi: 10.3233/JIFS-182709
  • Aktaş, H., Çağman, N., Soft sets and soft groups, Inf. Sci., 177 (2007), 2726–2735. https://doi.org/10.1016/j.ins.2006.12.008
  • Ulucay, V., Oztekin, O., Sahin, M., Olgun, N., Kargin, A., Soft representation of soft groups, New Trend Math. Sci., 4(2) (2016), 23. http://dx.doi.org/10.20852/ntmsci.2016217001
  • Feng, F., Jun, Y. B., Zhao, X., Soft semirings, Comput. Math. Appl., 56 (2008), 2621–2628. https://doi.org/10.1016/j.camwa.2008.05.011
  • Acar, U., Koyuncu, F., Tanay, B., Soft sets and soft rings, Comput. Math. Appl., 59 (2010), 3458-3463. https://doi.org/10.1016/j.camwa.2010.03.034
  • Uluçay, V., Şahin, M., Olgun, N., Soft normed rings, SpringerPlus, 5(1) (2016), 1–6. doi: 10.1186/s40064-016-3636-9
  • Atagün, A. O., Sezer, A. S., Soft substructures of rings fields and modules, Comput. Math. Appl., 61 (2011), 592–601. https://doi.org/10.1016/j.camwa.2010.12.005
  • Sezgin, A., Atagün, A. O., Aygün, E., A note on soft near-rings and idealistic soft near-rings, Filomat, 25 (2011), 53–68. doi: 10.2298/FIL1101053S
  • Ostadhadi-Dehkordi, S., Shum, K. P., Regular and strongly regular relations on soft hyperrings, Soft Comput., 23 (2019), 3253–3260. https://doi.org/10.1007/s00500-018-03711-8
  • Tahat, M. K., Sidky, F., Abo-Elhamayel, M., Soft topological soft groups and soft rings, Soft Comput., 22 (2018), 7143–7156. https://doi.org/10.1007/s00500-018-3026-z
  • Karaaslan, F., Some properties of AG*-groupoids and AG-bands under SI-product operation, J. Intell. Fuzzy Syst., 36 (2019), 231–239. doi: 10.3233/JIFS-181208
  • Yousafzaia, F., Khalaf, M. M., Alia, A., Arsham B., Saeidc, D., Non-associative ordered semigroups based on soft sets, Commun. Algebra, 47 (2019), 312–327. https://doi.org/10.1080/00927872.2018.1476524
  • Zhan, J., Dudek, W. A., Neggers, J., A new soft union set: characterizations of hemirings, Int. J. Mach. Learn. Cybern., 8 (2017), 525–535. https://doi.org/10.1007/s13042-015-0343-8
  • Atagün, A. O., Sezgin, A., Soft subnear-rings, soft ideals and soft N-subgroups of near-rings, Math. Sci. Lett., 7 (2018), 37–42. http://dx.doi.org/10.18576/msl/070106
  • Riaz, M., Naeem, K., Aslam, M., Afzal, D., Almahdi, F. A. A., Jamal, S. S., Multi-criteria group decision making with Pythagorean fuzzy soft topology, J. Intell. Fuzzy Syst., 39 (2020), 6703–6720. doi: 10.3233/JIFS-190854
  • Riaz, M., Naim, Ç., Zareef, I., Aslam, M., N-soft topology and its applications to multi-criteria group decision making, J. Intell. Fuzzy Syst., 36 (2019), 6521–6536. doi: 10.3233/JIFS-182919
  • Riaz, M., Tehreim, S. T., On bipolar fuzzy soft topology with decision-making, Soft Comput., 24 (2020), 18259–18272. https://doi.org/10.1007/s00500-020-05342-4
  • Sezer, A. S., Çağman, N., Atagün, A. O., Ali, M. I., Türkmen, E., Soft intersection semigroups, ideals and bi-ideals; a new application on semigroup theory I, Filomat, 29 (2015), 917–946. doi: 10.2298/FIL1505917S
  • Sezgin, A., Çağman, N., Çıtak, F., α-inclusions applied to group theory via soft set and logic, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68 (2019), 334–352. doi: 10.31801/cfsuasmas.420457
  • Feng, F., Li, C. X., Davvaz, B., Ali, M. I., Soft sets combined with fuzzy sets and rough sets: a tentative approach, Soft Comput., 14 (2010), 899–911. https://doi.org/10.1007/s00500-009-0465-6
  • Feng, F., Liu, X. Y., Leoreanu-Fotea, V., Jun, Y. B., Soft sets and soft rough sets, Inf. Sci., 181 (2011), 1125–1137. https://doi.org/10.1016/j.ins.2010.11.004
  • Atagün, A. O., Kamacı, H., Decompositions of soft sets and soft matrices with applications in group decision making, Scientia Iranica, in press (2021). doi:10.24200/SCI.2021.58119.5575.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Akın Osman Atagün 0000-0002-2131-9980

Hüseyin Kamacı 0000-0002-0429-6713

Publication Date December 30, 2022
Submission Date October 21, 2021
Acceptance Date May 18, 2022
Published in Issue Year 2022 Volume: 71 Issue: 4

Cite

APA Atagün, A. O., & Kamacı, H. (2022). Set-generated soft subrings of rings. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4), 993-1006. https://doi.org/10.31801/cfsuasmas.1013172
AMA Atagün AO, Kamacı H. Set-generated soft subrings of rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2022;71(4):993-1006. doi:10.31801/cfsuasmas.1013172
Chicago Atagün, Akın Osman, and Hüseyin Kamacı. “Set-Generated Soft Subrings of Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 4 (December 2022): 993-1006. https://doi.org/10.31801/cfsuasmas.1013172.
EndNote Atagün AO, Kamacı H (December 1, 2022) Set-generated soft subrings of rings. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 4 993–1006.
IEEE A. O. Atagün and H. Kamacı, “Set-generated soft subrings of rings”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 4, pp. 993–1006, 2022, doi: 10.31801/cfsuasmas.1013172.
ISNAD Atagün, Akın Osman - Kamacı, Hüseyin. “Set-Generated Soft Subrings of Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/4 (December2022), 993-1006. https://doi.org/10.31801/cfsuasmas.1013172.
JAMA Atagün AO, Kamacı H. Set-generated soft subrings of rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:993–1006.
MLA Atagün, Akın Osman and Hüseyin Kamacı. “Set-Generated Soft Subrings of Rings”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 4, 2022, pp. 993-1006, doi:10.31801/cfsuasmas.1013172.
Vancouver Atagün AO, Kamacı H. Set-generated soft subrings of rings. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(4):993-1006.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

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