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STRUCTURES OF INVOLUTION Γ-SEMIHYPERGROUPS

  • Yaqoob, Naveed (Department of Mathematics, College of Science Al-Zulfi, Majmaah University) ;
  • Tang, Jian (School of Mathematics and Statistics, Fuyang Normal University) ;
  • Chinram, Ronnason (Department of Mathematics and Statistics, Prince of Songkla University)
  • Received : 2017.10.28
  • Accepted : 2018.01.10
  • Published : 2018.03.25

Abstract

In this paper, structure of involution ${\Gamma}$-semihypergroup is introduced and some theorems about this concept are stated and proved. The concept of ${\Gamma}$-hyperideal in involution ${\Gamma}$-semihypergroup is defined and some of their properties are studied. Some results on regular ${\Gamma}^*$-semihypergroups and fuzzy ${\Gamma}^*$-semihypergroups are also provided.

Keywords

${\Gamma}$-semihypergroup;${\Gamma}$-hyperideal;Involution

References

  1. S. Abdullah, K. Hila and M. Aslam, On bi-${\Gamma}$-hyperideals of ${\Gamma}$-semihypergroups, U.P.B. Scientific Bulletin. Series A, 74(4) (2012) 79-90.
  2. U.A. Aburawash, Semiprime involution rings and chain conditions, Mathematica Japonica, 37 (1992) 987-994
  3. U.A. Aburawash, On *-simple involution rings with minimal *-bi-ideals, Studia Scientiarum Mathematicarum Hungarica, 32 (1996) 455-458.
  4. U.A. Aburawash and M.A. Shatila, On group rings with involution, International Journal of Basic and Applied Sciences, 13(5) (2013) 28-31.
  5. S.M. Anvariyeh, S. Mirvakili and B. Davvaz, On ${\Gamma}$-hyperideals in ${\Gamma}$-semihypergroups, Carpathian Journal of Mathematics, 26 (2010) 11-23.
  6. W.E. Baxter, On rings with proper involution, Pacific Journal of Mathematics, 27(1) (1968) 1-12. https://doi.org/10.2140/pjm.1968.27.1
  7. K.I. Beidar and R. Wiegandt, Rings with involution and chain conditions, Journal of Pure and Applied Algebra, 87(3) (1993) 205-220. https://doi.org/10.1016/0022-4049(93)90109-7
  8. P. Civin and B. Yood, Involutions on Banach algebras, Pacific Journal of Mathematics, 9 (1959) 415-436. https://doi.org/10.2140/pjm.1959.9.415
  9. P. Corsini, M. Shabir and T. Mahmood, Semisipmle semihypergroups in terms of hyperideals and fuzzy hyperideals, Iranian Journal of Fuzzy Systems, 8(1) (2011) 95-111.
  10. P. Corsini and V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publications, (2003).
  11. B. Davvaz and V. Leoreanu-Fotea, Structures of Fuzzy ${\Gamma}$-hyperideals in ${\Gamma}$-semihypergroups, Journal of Multiple-Valued Logic and Soft Computing, 19 (2012) 519-535.
  12. B. Davvaz, Fuzzy hyperideals in semihypergroups, Italian Journal of Pure and Applied Mathematics, 8 (2000) 67-74.
  13. B. Davvaz, Intuitionistic hyperideals of semihypergroups, Bulletin of the Malaysian Mathematical Sciences Society, 29(1) (2006) 203-207.
  14. M.P. Drazin, Natural structures on semigroups with involution, Bulletin of the American Mathematical Society, 84(1) (1978) 139-141. https://doi.org/10.1090/S0002-9904-1978-14442-5
  15. M.P. Drazin, Regular semigroups with involution, Proceedings of the Symposium on Regular Semigroups, (DeKalb, 1979), 29-46.
  16. B.A. Ersoy and B. Davvaz, Atanassov's intuitionistic fuzzy ${\Gamma}$-hyperideals of ${\Gamma}$-semihypergroups, Journal of Intelligent and Fuzzy Systems, 25 (2013) 463-470.
  17. X. Feng, J. Tang, B. Davvaz and Y. Luo, A novel study on fuzzy ideals and fuzzy filters of ordered *-semigroups, Journal of Intelligent and Fuzzy Systems, 33(1) (2017) 423-431. https://doi.org/10.3233/JIFS-161740
  18. D.J. Foulis, Involution semigroups, Ph.D. Thesis, Tulane University, New Orleans, (1958).
  19. D. Heidari, S.O. Dehkordi and B. Davvaz, ${\Gamma}$-Semihypergroups and their properties, U.P.B. Scientific Bulletin. Series A, 72 (2010) 197-210.
  20. I.N. Herstein, Special simple rings with involution, Journal of Algebra, 6 (1967) 369-375. https://doi.org/10.1016/0021-8693(67)90089-0
  21. I.N. Herstein, Ring with involution, University of Chicago Press, Chicago, (1976).
  22. K. Hila, B. Davvaz and J. Dine, Study on the structure of ${\Gamma}$-semihypergroup, Communications in Algebra, 40(8) (2012) 2932-2948. https://doi.org/10.1080/00927872.2011.587855
  23. K. Hila and A. Gilani, On fuzzy subsets in ${\Gamma}$-semihypergroup through left operator semihypergroup, International Conference on Applied Analysis and Algebra (ICAAA2011), 29-30 June and 1-2 July 2011 in Istanbul, Turkey.
  24. T.P. Lim, Some classes of rings with involution satisfying the standard polynomial of degree 4, Pacific Journal of Mathematics, 85(1) (1979) 125-130. https://doi.org/10.2140/pjm.1979.85.125
  25. F. Marty, Sur une generalization de la notion de groupe, 8iem congres Math. Scandinaves, Stockholm, (1934) 45-49.
  26. D.I.C. Mendes, A note on involution rings, Miskolc Mathematical Notes, 10(2) (2009) 155-162.
  27. L. Oukhtite, On Jordan ideals and derivations in rings with involution, Commentationes Mathematicae Universitatis Caroline, 3(51) (2010) 389-395.
  28. N.R. Reilly, A class of regular *-semigroups, Semigroup Forum, 18(1) (1979) 385-386. https://doi.org/10.1007/BF02574203
  29. H.E. Scheiblich and T.E. Nordahl, Regular *-semigroups, Semigroup Forum, 16(1) (1978) 369-377. https://doi.org/10.1007/BF02194636
  30. M.K. Sen and N.K. Saha, On ${\Gamma}$-semigroup I, Bulletin of Calcutta Mathematical Society, 78 (1986) 180-186.
  31. L.B. Small, Mappings on simple rings with involution, Journal of Algebra, 13(1) (1969) 119-136. https://doi.org/10.1016/0021-8693(69)90009-X
  32. C.-Y. Wu, On intra-regular ordered *-semigroups, Thai Journal of Mathematics, 12(1) (2014) 15-24.