Advertisement

Afrika Matematika

, Volume 29, Issue 1–2, pp 97–114 | Cite as

Rough approximations in non-commutative residuated lattices

  • Mahmood Bakhshi
  • Mahdi Izanlou
Article
  • 65 Downloads

Abstract

Based on Pawlak’s rough set theory, we study and investigate the roughness in non-commutative residuated lattices, which are generalizations of non-commutative fuzzy structures such as MV-algebras and BL-algebras. We give many theorems and examples to describe the rough approximations. Also, to investigate the properties of roughness of subsets (and of course filters) more closely, we consider some different kinds of filters such as Boolean filters and prime filters. Especially, we prove that with respect to some certain filters, the obtained approximations form a Boolean algebra or a pseudo MTL-algebra.

Keywords

Algebras of fuzzy logics Residuated lattice Boolean algebra Rough approximation 

Mathematics Subject Classification

03G25 08A72 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments.

References

  1. 1.
    Bakhshi, M.: Boolean filters and prime filters of non-commutative residuated lattices. In: Extended Abstract of the 42nd Annual Iraninan Mathematics Conference, Sep. 5–8, 2011, Vali-e-Asr University of Rafsanjan, IranGoogle Scholar
  2. 2.
    Biswas, R., Nanda, S.: Rough groups and rough subgroups. Bull. Pol. Acad. Sci. Math. 42(3), 251–254 (1994)MathSciNetMATHGoogle Scholar
  3. 3.
    Ciungu, L.C.: Classes of residuated lattices. Ann. Univ. Craiova Math. Comput. Sci. Ser. 33, 189–207 (2006)MathSciNetMATHGoogle Scholar
  4. 4.
    Comer, S.D.: On connections between information systems, rough sets and algebraic logic. Algebraic Methods Logic Comput. Sci. 28, 117–124 (1993)MathSciNetMATHGoogle Scholar
  5. 5.
    Davvaz, B.: Roughness in rings. Inf. Sci. 164, 147–163 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Davvaz, B.: Roughness based on fuzzy ideals. Inf. Sci. 176, 2417–2437 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dilworth, R.P.: Non-commutative residuated lattices. Trans. Am. Math. Soc. 46(3), 426–444 (1939)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Estaji, A.A., Hooshmandasl, M.R., Davvaz, B.: Rough set theory applied to lattice theory. Inf. Sci. 200, 108–122 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iwinski, T.B.: Algebraic approach to rough sets. Bull. Pol. Acad. Sci. 35, 673–683 (1987)MathSciNetMATHGoogle Scholar
  10. 10.
    Jun, Y.B.: Roughness of ideals in BCK-algebras. Sci. Math. Jpn. 7, 115–119 (2002)Google Scholar
  11. 11.
    Kazanci, O., Davvaz, B.: On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. Inf. Sci. 178, 1343–1354 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kuroki, N.: Rough ideals in semigroups. Inf. Sci. 100, 139–163 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kuroki, N., Mordeson, J.N.: Structure of rough sets and rough groups. J. Fuzzy Math. 5, 183–191 (1997)MathSciNetMATHGoogle Scholar
  14. 14.
    Lianzhen, L., Kaitai, K.: Boolean filters and positive implicative filters of residuated lattices. Inf. Sci. 177, 5726–5738 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–357 (1982)CrossRefMATHGoogle Scholar
  16. 16.
    Pomykala, J., Pomykala, J.A.: The Stone algebra of rough sets. Bull. Pol. Acad. Sci. Math. 36, 495–508 (1988)MathSciNetMATHGoogle Scholar
  17. 17.
    Rachu̇ek, J., Šalounová, D.: Roughness in residuated lattices. Advances in computational intelligence. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) Proc. 14th Int. Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 596–603. IPMU 2012, Catania, Italy (2012)Google Scholar
  18. 18.
    Rasouli, S., Davvaz, B.: Roughness in MV-algebras. Inf. Sci. 180, 737–747 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Torkzadeh, L., Ghorbani, S.: Rough filters in BL-algebras. Int. J. Math. Math. Sci. doi: 10.1155/2011/474375 (2011)
  20. 20.
    Ward, M., Dilworth, R.P.: Residuated lattices. Trans. Am. Math. Soc. 45, 335–354 (1939)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Xiao, Q.M., Zhang, Z.L.: Rough prime ideals and rough fuzzy prime ideals in semigroups. Inf. Sci. 176, 725–733 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhu, Y., Xu, Y.: On filter theory of residuated lattices. Inf. Sci. 180, 3614–3632 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BojnordBojnordIran

Personalised recommendations