International Journal of Analysis and Applications ISSN 2291-8639 (Online)

Research Article

Fuzziness and Roughness in Hyperquantales

  • Raees Khan1*
  • Maslina Darus2
  • Muhammad Farooq3
  • Asghar Khan3
  • Nasir Khan3
  1. FATA University, Darra Adam Khel N.M.D. Kohat, KP, Pakistan
  2. Bacha Khan University, Charsadda, KP, Pakistan
  3. Abdul Wali Khan University, Mardan, KP, Pakistan

* Corresponding author

Abstract

Theories of fuzzy set and rough set are powerful mathematical tools for modelling various types of uncertainty. In this paper, we introduce the notions of bi-hyperideal, fuzzy bi-hyperideals of hyperquantales and their related properties is given. Furthermore we introduce the notion of generalized rough fuzzy bi-hyperideals. Moreover, we will describe the set-valued homomorphism and strong set-valued homomorphism of hyperquantales and some related properties will be study.

How to Cite

R. Khan, M. Darus, M. Farooq, A. Khan, N. Khan, Fuzziness and Roughness in Hyperquantales, Int. J. Anal. Appl. 18 (2020), 998–1014. https://doi.org/10.28924/2291-8639-18-2020-998

BibTeX
@article{Khan2020,
  author  = {Khan, Raees and Darus, Maslina and Farooq, Muhammad and Khan, Asghar and Khan, Nasir},
  title   = {Fuzziness and Roughness in Hyperquantales},
  journal = {Int. J. Anal. Appl.},
  volume  = {18},
  year    = {2020},
  pages   = {998--1014},
  doi     = {10.28924/2291-8639-18-2020-998}
}

References

  • [1] F. Marty, Sur une generalisation de la notion de group, in Proceedings of the 8th Congress des Mathematiciens Scandinaves, Stockholm, 1934, pp. 45-49.
  • [2] J. Tang, B. Davvaz and Y. F. Luo, A study on fuzzy interior hyperideals in ordered semihypergroups, Ital. J. Pure Appl. Math. 36 (2016), 125-146.
  • [3] J. Tang, A. Khan and Y. F. Luo, Characterization of semisimple ordered semihypergroups in terms of fuzzy hyperideals, J. Intell. Fuzzy Syst. 30 (2016), 1735-1753.
  • [4] B. Davvaz, A. Khan and M. Farooq, Int-soft structures applied to ordered semihypergroups, Le Matematiche, LXXIII (2018), 235-259.
  • [5] A. Khan, M. Farooq and H. U. Khan, Uni-soft hyperideals of ordered semihypergroups, J. Intell. Fuzzy Syst. 35 (2018), 4557-4571.
  • [6] B. Davvaz, A. Khan and M. Khan, Atanassov's intuitionistic fuzzy set theory applied to quantales, Novi Sad J. Math. 47 (2) (2017), 47-61.
  • [7] B. B. N. Koguep, On fuzzy Ideals of hyperlattice, Int. J. Algebra, 2 (15) (2008), 739-750.
  • [8] P. Corsini, V. Leoreanu, Applications of hyperstructure theory, Kluwer Academic Publishers, Dordrecht, 2003.
  • [9] J. Zhan, B. Davvaz, P. K. Shun, Probability n-ary bypergroups. Inform. Sci. 180 (2010), 1159-1166.
  • [10] J. Zhan, C. Irina, Γ-hypermodules: isomorphism theorems and regular relations. U. P. B. Sci Bull. Ser. A. 73 (2011), 71-78.
  • [11] J. Zhan, B. Davvaz, P. K. Shun., A new view of fuzzy hyperrings. Inform. Sci. 178 (2008), 425-438.
  • [12] M. Konstantinidou and J. Mittas, An introduction to the theory of hyperlattices, Math. Balkanica, 7 (1977), 187-193.
  • [13] S. Rasouli and B. Davvaz, Lattices derived from hyperlattices. Commun. Algebra, 38 (8) (2010), 2720-2737.
  • [14] S. Rasouli and B. Davvaz, Construction and spectral topology on hyperlattices, Mediterr. J. Math. 7 (2) (2010), 249-262.
  • [15] Z. Pawlak, Information systems-theoretical foundations, Inform. Syst. 6 (1981), 205-218.
  • [16] Z. Pawlak, Rough sets, Int. J. Comput. Inform. Sci. 11 (1982), 341-356.
  • [17] J. Jarvinen, Lattice Theory for Rough Sets, Transactions on Rough Sets VI, LNCS 4374, Springer-Verlag, Berlin Heidelberg, 2007 (pp. 400-498).
  • [18] G.L. Liu and W. Zhu, The algebraic structures of generalized rough set theory, Inform. Sci. 178 (2008), 4105-4113.
  • [19] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353.
  • [20] C. J. Mulvey, Supplemento ai Rendiconti del Circolo Matematico di Palermo II 12 (1986), 99-104.
  • [21] K. Y. Wang, Prime radical theorem in quantales, Fuzzy Syst. Math. 25 (2) (2011), 60-64 (in Chinese).
  • [22] S. Q. Wang and B. Zhao, Ideals of quantales, J. Shaanxi Normal Univ. (Nat. Sci. Ed.) 31 (4) (2003), 7-10 (in Chinese).
  • [23] S. Q. Wang and B. Zhao, Prime ideal and weakly prime ideal of the quantale, Fuzzy Syst. Math. 19 (1) (2005), 78-81 (in Chinese).
  • [24] L. Y. Yang, L.S. Xu, Roughness in quantales, Inform. Sci. 220 (2013), 568-579.
  • [25] Q. Luo and G. Wang, Roughness and fuzziness in quantales, Inform. Sci. 271 (2014), 14-30.
  • [26] M. Farooq, T. Mahmood, A. Khan, M. Izhar and B. Davvaz, Fuzzy hyperideals of hyperquantale, J. Intell. Fuzzy Syst. 36 (2019), 5605-5615.
  • [27] S. Yamak, O. Kazanci, and B. Davvaz, Generalized lower and upper approximations in a ring, Inform. Sci. 180 (9) (2010), 1759-1768.
  • [28] A. K. Estaji and F. Bayati, On Rough Sets and Hyperlattices, Ratio Math. 34 (2018), 15-33.
  • [29] S. M. Qurashi and M. Shabir, Generalized Rough Fuzzy Ideals in Quantales, Discrete Dyn. Nat. Soc. 2018 (2018), Article ID 1085201.
  • [30] S. M. Qurashi and M. Shabir, Generalized approximations of (∈, ∈ ∨q)-fuzzy ideals in quantales, Comput. Appl. Math. 37 (2018), 6821-6837.