Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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GENERALIZED((ξ, ζ)-) SOFT INTERIOR Γ-HYPERIDEALS OF Γ-SEMIHYPERGROUPS

  • Abbasi, M.Y. (Department of Mathematics, Jamia Millia Islamia) ;
  • Basar, A. (Department of Mathematics, Jamia Millia Islamia) ;
  • Talee, A.F. (Department of Mathematics, Jamia Millia Islamia) ;
  • Khan, S.A. (Department of Mathematics, Jamia Millia Islamia, Mohammad Ali Jauhar)
  • Received : 2017.10.24
  • Accepted : 2017.12.08
  • Published : 2018.03.25
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Abstract

In this paper, we introduce the notion of ($\xi$, ${\zeta}$)-soft ${\Gamma}$-hyperideals and ($\xi$, ${\zeta}$)-soft interior ${\Gamma}$-hyperideals of ${\Gamma}$-semihypergroups by a new approach called soft intersection (briefly, S. I.). It is proved that in regular ${\Gamma}$-semihypergroups the ($\xi$, ${\zeta}$)-soft ${\Gamma}$-hyperideals and the ($\xi$, ${\zeta}$)-soft interior ${\Gamma}$-hyperideals coincide. Further, we introduce the concept of ($\xi$, ${\zeta}$)-soft simple ${\Gamma}$-semihypergroup and characterize the simple ${\Gamma}$-semihypergroups in terms of ($\xi$, ${\zeta}$)-soft ${\Gamma}$-hyperideals and ($\xi$, ${\zeta}$)-soft interior ${\Gamma}$-hyperideals.

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References

  1. M.Y. Abbasi, B. Davvaz, S.A. Khan and A.F. Talee, On Some Certain Characterizations of Soft ${\Gamma}$Hyperideals in ${\Gamma}$-Semihypergroups by a New Approach (Communicated).
  2. H. Aktas and N. Cagman, Soft sets and soft groups, Inform. Sci., 177 (2007), 2726-2735. https://doi.org/10.1016/j.ins.2006.12.008
  3. S. M. Anvariyeh, S. Mirvakili and B. Davvaz, On ${\Gamma}$-hyperideals in ${\Gamma}$-semihypergroups, Carpathian J. Math., 26(1) (2010), 11-23.
  4. S. M. Anvariyeh, S. Mirvakili and B. Davvaz, Pawlak's approximations in ${\Gamma}$-semihypergroups, Computers and Mathematics with Applications, 60 (2010), 45-53. https://doi.org/10.1016/j.camwa.2010.04.028
  5. S. M. Anvariyeh, S. Mirvakili, O. Kazanci and B. Davvaz, Algebraic hyperstructures of soft sets associated to semihypergroups, Southeast Asian Bulletin of Mathematics, 35 (2011), 911-925.
  6. M. Aslam, S. Abdullah, B. Davvaz and N. Yaqoob, Rough M-hypersystems and fuzzy M-hypersystems in ${\Gamma}$-semihypergroups, Neural Computing & Applications, 21 (2012), 281-287.
  7. N. Cagman and S. Enginoglu, Soft set theory and uni-int decision making, Eur. J. Op. Res., 207 (2010), 848-855. https://doi.org/10.1016/j.ejor.2010.05.004
  8. N. Cagman, F. Citak and H. Aktas, Soft int-group and its applications to group theory, Neural Comput. Appl., 21 (2012), 151-158. https://doi.org/10.1007/s00521-011-0752-x
  9. P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer, Dordrecht, (2003).
  10. P. Corsini, Prolegomena of hypergroup theory, Aviani editor, Second edition, (1993).
  11. B. Davvaz and V.L. Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harber, Fla, USA, 115 (2007).
  12. B. Davvaz and V.L. Fotea, Structures of fuzzy ${\Gamma}$-hyperideals in ${\Gamma}$-semihypergroups, Multiple-Valued Logic and Soft Computing, 19 (2012), 519-535.
  13. B. Davvaz and V.L. Fotea, Triangular fuzzy sub ${\Gamma}$-semihypergroups in ${\Gamma}$-semihypergroups, Kuwait J. Sci., 40(1) (2014), 93-106.
  14. B.A. Ersoy, Y. Saricaoglu, M. Yenigun and B. Davvaz, On fuzzy interior ${\Gamma}$-hyperideals of ${\Gamma}$-semihypergroups, Utilitas Mathematica, 88 (2012), 157-170.
  15. M. Farooq, A. Khan and B. Davvaz, Characterizations of ordered semihyper-groups by the properties of their intersectional-soft generalized bi-hyperideals, Soft Comput, DOI 10.1007/s00500-017-2550-6, (2017).
  16. F. Feng, Y.B. Jun and X. Zhao, Soft semirings, Comput. Math. Appl., 56 (2008), 2621-2628. https://doi.org/10.1016/j.camwa.2008.05.011
  17. Y. Feng and P. Corsini, $({\lambda},\;{\mu})$-fuzzy interior ideals of ordered ${\Gamma}$-semigroups, Algebra and Discrete Mathematics, 16(1) (2013), 61-70.
  18. A. Hasankhani, Ideals in a semihypergroup and Green's relations, Ratio Mathematica, 13 (1999), 29-36.
  19. D. Heidari, S.O. Dehkordi and B. Davvaz, ${\Gamma}$-Semihypergroups and their properties, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 72(1) (2010), 195-208.
  20. D. Heidari and B. Davvaz, ${\Gamma}$-hypergroups and ${\Gamma}$-Semihypergroups associated to binary relations, Iranian Journal of Science and Technology, Transaction A, A2, (2011), 69-80.
  21. K. Hila, B. Davvaz and J. Dine, Study on the Structure of ${\Gamma}$-Semihypergroups, Communications in Algebra, 40 (2012), 2932-2948. https://doi.org/10.1080/00927872.2011.587855
  22. P.K. Maji, R. Biswas and A.R. Roy, Soft set theory, Comput. Math. Appl., 45 (2003), 555-562. https://doi.org/10.1016/S0898-1221(03)00016-6
  23. F. Marty, Sur une generalization de la notion de group, 8th Congres Math. Scandinaves Stockholm, (1934), 45-49.
  24. D. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19-31.
  25. S. Naz and M. Shabir, On soft semihypergroups, Journal of Intelligent and Fuzzy Systems, 26 (2014), 2203-2213.
  26. S.K. Sardar, B. Davvaz and S.K. Majumdera, A study on fuzzy interior ideals of ${\Gamma}$-semigroups, Computers and Mathematics with Applications, 60 (2010), 90-94. https://doi.org/10.1016/j.camwa.2010.04.033
  27. M.K. Sen, On ${\Gamma}$-semigroups, Algebra and its applications, Int. Symp., New Delhi, (1981)
  28. M.K. Sen, On ${\gamma}$-semigroups, Algebra and its applications, Lecture Notes in Pure and Applied Mathematics 91, Decker, New York, (1984), 301-308.
  29. A. Sezgin and A.O. Atagun, On operations of soft sets, Comput. Math. Appl., 61 (2011), 1457-1467. https://doi.org/10.1016/j.camwa.2011.01.018
  30. A. Sezgin, N. Cagman and A.O. Atagun, Soft intersection interior ideals, quasiideals and generalized bi-ideals; A New Approach to Semigroup Theory II, J. of Mult.-Valued Logic and Soft Computing, 0 (2013), 1-47.
  31. A. Sezgin, N. Cagman, A.O. Atagun and M. Ali, Soft intersection semigroups,ideals and bi-ideals; a new approach to semigroup theory, Filomat, 29(5) (2015), 917-946 https://doi.org/10.2298/FIL1505917S
  32. J. Tang, B. Davvaz and Y. Luo, A study on fuzzy interior hyperideals in ordered semihypergroups, Italian Journal of Pure and Applied Mathematics, 36 (2016), 125-146.
  33. J. Tang, B. Davvaz, X.Y. Xie and N. Yaqoob, On fuzzy interior ${\Gamma}$-hyperideals in ordered ${\Gamma}$-semihypergroups, Journal of Intelligent and Fuzzy Systems, 32 (2017), 2447-2460. https://doi.org/10.3233/JIFS-16431
  34. N. Tipachot and B. Pibaljommee, Fuzzy interior hyperideals in ordered semihypergroups, Italian Journal of Pure and Applied Mathematics, 36 (2016), 859-870.
  35. N. Yaqoob, M. Aslam, B. Davvaz and A. Ghareeb, Structures of bipolar fuzzy ${\Gamma}$-hyperideals in ${\Gamma}$-semihypergroups, Journal of Intelligent and Fuzzy Systems, 27 (2014), 3015-3032.
  36. N. Yaqoob, M. Aslam, K. Hila and B. Davvaz, Rough prime bi-G-hyperideals and fuzzy prime bi-G-hyperideals of G-semihypergroups, Filomat, 31(13), (2017), 4167-4183. https://doi.org/10.2298/FIL1713167Y
  37. N. Yaqoob and M. Aslam, Generalized rough approximations in G-semihypergroups, Journal of Intelligent and Fuzzy Systems, 27(5) (2014), 2445-2452.
  38. N. Yaqoob, M. Aslam and Faisal, On soft G-hyperideals over left almost G-semihypergroups, Journal of Advanced Research in Dynamical and Control Systems, 4(1) (2012), 1-12.