Using the notion of generalized fuzzy sets, we introduce the notions of generalized fuzzy hyperideals, generalized fuzzy bi-hyperideals, and generalized fuzzy normal bi-hyperideals in an ordered nonassociative and non-commutative algebraic structure, namely an ordered LA-semihypergroup, and we characterize these hyperideals. We provide some results related to the images and preimages of generalized fuzzy hyperideals in ordered LA-semihypergroups.
1. Marty, F. Sur une generalization de la notion de groupe. In 8iem Congres Mathématiciens Scandinaves. Stockholm., 1934, 45–49.
2. Corsini, P. and Leoreanu, V. Applications of Hyperstructure Theory. Kluwer Academic Publications, 2003.
https://doi.org/10.1007/978-1-4757-3714-1
3. Davvaz, D. and Fotea, V. L. Hyperring Theory and Applications. International Academic Press, USA, 2007.
4. Bonansinga, P. and Corsini, P. On semihypergroup and hypergroup homomorphisms. Boll. Un. Mat. Ital., 1982, 6, 717–727.
5. Corsini, P. and Cristea, I. Fuzzy sets and non complete 1-hypergroups. An. Sti. U. Ovid. Co-Mat., 2005, 13, 27–54.
6. Davvaz, B. Some results on congruences on semihypergroups. Bull. Malays. Math. Sci. Soc., 2000, 23, 53–58.
7. Hasankhani, A. Ideals in a semihypergroup and Green’s relations. Ratio Math., 1999, 13, 29–36.
8. Hila, K. and Dine, J. On hyperideals in left almost semihypergroups. ISRN Algebra, 2011, Article ID 953124, 8 pages.
9. Yaqoob, N., Corsini, P., and Yousafzai, F. On intra-regular left almost semihypergroups with pure left identity. J. Math., 2013, Article ID 510790, 10 pages.
10. Yousafzai, F. and Corsini, P. Some characterization problems in LA-semihypergroups. J. Algebra, Numb. Th. Adv. Appl., 2013, 10, 41–55.
11. Heidari, D. and Davvaz, B. On ordered hyperstructures. U.P.B. Sci. Bull. Series A, 2011, 73, 85–96.
12. Yaqoob, N. and Gulistan, M. Partially ordered left almost semihypergroups. J. Egyptian Math. Soc., 2015, 23, 231–235.
https://doi.org/10.1016/j.joems.2014.05.012
13. Zadeh, L. A. Fuzzy sets. Inform. Control, 1965, 8, 338–353.
https://doi.org/10.1016/S0019-9958(65)90241-X
14. Murali, V. Fuzzy points of equivalent fuzzy subsets. Inform. Sci., 2004, 158, 277–288.
https://doi.org/10.1016/j.ins.2003.07.008
15. Pu, P. M. and Liu, Y. M. Fuzzy topology I, neighborhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appl., 1980, 76, 571–599.
https://doi.org/10.1016/0022-247X(80)90048-7
16. Bhakat, S. K. and Das, P. (2;2 _q)-fuzzy subgroups. Fuzzy Sets Syst., 1996, 80, 359–368.
https://doi.org/10.1016/0165-0114(95)00157-3
17. Pibaljommee, B., Wannatong, K., and Davvaz, B. An investigation on fuzzy hyperideals of ordered semihypergroups. Quasigroups Relat. Syst., 2015, 23, 297–308.
18. Tang, J., Khan, A., and Luo, Y. F. Characterizations of semisimple ordered semihypergroups in terms of fuzzy hyperideals. J. Intell. Fuzzy Syst., 2016, 30, 1735–1753.
https://doi.org/10.3233/IFS-151884
19. Azhar, M., Gulistan, M., Yaqoob, N., and Kadry, S. On fuzzy ordered LA-semihypergroups. Int. J. Anal. Appl., 2018, 16, 276–289.
20. Azhar, M., Yaqoob, N., Gulistan, M., and Khalaf, M. On (2;2 _qk)-fuzzy hyperideals in ordered LA-semihypergroups. Disc. Dyn. Nat. Soc., 2018, Article ID 9494072, 13 pages.
21. Shabir, M. and Mahmood, T. Semihypergroups characterized by (2g ;2g _qd )-fuzzy hyperideals. J. Intell. Fuzzy Syst., 2015, 28, 2667–2678.
https://doi.org/10.3233/IFS-151544
22. Shabir, M., Jun, Y. B., and Nawaz, Y. Semigroups characterized by (2g ;2g _qk)-fuzzy ideals. Comput. Math. Appl., 2010, 60, 1473–1493.
https://doi.org/10.1016/j.camwa.2010.06.030
23. Rehman, N. and Shabir, M. Some characterizations of ternary semigroups by the properties of their (2g ;2g _qd )-fuzzy ideals. J. Intell. Fuzzy Syst., 2014, 26, 2107–2117.
