Computational and Applied Mathematics

, Volume 37, Issue 4, pp 5013–5033 | Cite as

Generalized roughness in fuzzy filters and fuzzy ideals with thresholds in ordered semigroups

  • Tahir Mahmood
  • Muhammad Irfan Ali
  • Azmat Hussain


In the present paper, concept of roughness for fuzzy filters with thresholds \(\left( u_{1},u_{2}\right) \) in ordered semigroups is introduced. Then, this concept is extended to fuzzy bi-filters with thresholds and fuzzy quasi-filters with thresholds. Further approximations of fuzzy ideals with thresholds, fuzzy bi-ideals with thresholds, and fuzzy interior ideals with thresholds are studied. Moreover, this concept is applied to study approximations of fuzzy quasi-ideals with thresholds and semiprime fuzzy quasi-ideals with thresholds.


Rough sets Fuzzy set Ordered Semigroups Approximation of fuzzy filters with thresholds Approximation of fuzzy ideals with thresholds 

Mathematics Subject Classification

08A72 34C41 06F05 


  1. Banerjee M, Pal SK (1996) Roughness of a fuzzy set. Inf Sci 93:235–246MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bhakat SK, Das P (1996) \(\left( \in,\in \vee q\right) \)-fuzzy subgroups. Fuzzy Set Syst 80:359–368CrossRefzbMATHGoogle Scholar
  3. Biswas R, Nanda S (1994) Rough groups and rough subgroups. Bull Polish Acad Sci Math 42:251MathSciNetzbMATHGoogle Scholar
  4. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in rough set theory. Inf Sci 107:149–167MathSciNetCrossRefzbMATHGoogle Scholar
  5. Chakrabarty K, Biswas R, Nanda S (2000) Fuzziness in rough sets. Fuzzy Sets Syst 110:247–251MathSciNetCrossRefzbMATHGoogle Scholar
  6. Davvaz B (2004) Roughness in rings. Inf Sci 164:147–163MathSciNetCrossRefzbMATHGoogle Scholar
  7. Davvaz B (2008) A short note on algebra T-rough sets. Inf Sci 178:3247–3252CrossRefzbMATHGoogle Scholar
  8. Davvaz B, Khan A (2012) Generalized fuzzy filter in ordered semigroup. Iran J Sci Technol 36:77–86MathSciNetGoogle Scholar
  9. Davvaz B, Mahdavipour M (2006) Roughness in modules. Inf Sci 176:3658–3674MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–208CrossRefzbMATHGoogle Scholar
  11. Hosseini SB, Jafarzadeh N, Gholami A (2012) Some results on T-rough (prime, primary) ideal and T-rough fuzzy (prime, primary) ideal on commutative rings. Int J Contemp Math Sci 7:337–350MathSciNetzbMATHGoogle Scholar
  12. Hosseini SB, Jafarzadeh N, Gholami A (2012) T-rough ideal and T-rough fuzzy ideal in a semigroup. Adv Mater Res 433:4915–4919CrossRefzbMATHGoogle Scholar
  13. Iwinski TB (1987) Algebraic approach to rough sets. Bull Polish Acad 35:673–683MathSciNetzbMATHGoogle Scholar
  14. Jirojkul C, Chinram R (2009) Fuzzy quasi-ideal subsets and fuzzy quasi-filters of ordered semigroup. Int J Pure Appl Math 52:611–617MathSciNetzbMATHGoogle Scholar
  15. Jun YB (2003) Roughness of ideals in BCK-algebra. Sci Math Jpn 57:165–169MathSciNetzbMATHGoogle Scholar
  16. Jun YB, Ahn SS, Khan A (2014) Bi-filters of ordered semigroups related to fuzzy points. Appl Math Sci 8:1011–1029MathSciNetGoogle Scholar
  17. Kazanci O, Yamak S (2008) Generalized fuzzy bi-ideals of semigroup. Soft Comput 12:1119–1124CrossRefzbMATHGoogle Scholar
  18. Kazanci O, Davvaz B (2008) On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings. Inf Sci 178:1343–1354MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kehayopulu N, Tsingelis M (1999) A note on fuzzy sets in semigroups. Sci Math 2:411–413MathSciNetzbMATHGoogle Scholar
  20. Kehayopulu N, Tsingelis M (2002) Fuzzy sets in ordered groupoids. Semigroup Forum 65:128–132MathSciNetCrossRefzbMATHGoogle Scholar
  21. Kehayopulu N, Tsingelis M (2005) Fuzzy bi-ideals in ordered semigroups. Inf Sci 171:13–28MathSciNetCrossRefzbMATHGoogle Scholar
  22. Kehayopulu N, Tsingelis M (2006) Fuzzy interior ideals in ordered semigroups. Lobachevskii J Math 21:65–71MathSciNetzbMATHGoogle Scholar
  23. Kehayopulu N, Tsingelis M (2009) Fuzzy right, left, quasi-ideals, bi-ideals in ordered semigroups. Lobachevskii J Math 30:17–22MathSciNetCrossRefzbMATHGoogle Scholar
  24. Kuroki N (1979) Fuzzy bi-ideals in semigroups. Commentarii Mathematici Universitatis Sancti Pauli 28:17–21MathSciNetzbMATHGoogle Scholar
  25. Kuroki N (1981) On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets Syst 5:203–215MathSciNetCrossRefzbMATHGoogle Scholar
  26. Kuroki N (1991) On fuzzy semigroups. Inf Sci 53:203–236CrossRefzbMATHGoogle Scholar
  27. Kuroki N (1993) Fuzzy semiprime quasi-ideals in semigroups. Inf Sci 75:201–211MathSciNetCrossRefzbMATHGoogle Scholar
  28. Kuroki N (1997) Rough ideals in semigroups. Inf Sci 100:139–163MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ma X, Zhan J, Jun YB (2009) On \((\in,\in \vee q)\) fuzzy filters of \(R_{0}\)-algebras. Math Logic Quart 55:493–508CrossRefzbMATHGoogle Scholar
  30. Manzoor R, Khan A, Yousafzai F, Amjad V (2013) Fuzzy quasi-ideals with thresholds \(\left( \alpha,\beta \right] \) in ordered semigroups. Int J Algebra Stat 2:72–82CrossRefGoogle Scholar
  31. Pawlak Z (1982) Rough sets. Int J Comput Sci 11:341–356zbMATHGoogle Scholar
  32. Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35:512–517MathSciNetCrossRefzbMATHGoogle Scholar
  33. Shabir M, Nawaz Y, Aslam M (2011) Semigroups characterized by the properties of their fuzzy ideals with thresholds. World Appl Sci J 14:1851–1865Google Scholar
  34. Siripitukdet M, Ruanon A (2013) Fuzzy interior ideals with thresholds (s, t] in ordered semigroups. Thai J Math 11:371–382MathSciNetzbMATHGoogle Scholar
  35. Yao YY (1998) Constructive and algebraic methods of the theory of rough sets. Inf Sci 109:21–47CrossRefzbMATHGoogle Scholar
  36. Yuan X, Zhang C, Ren Y (2003) Generalized fuzzy groups and many-valued implications. Fuzzy Sets Syst 138:205–211MathSciNetCrossRefzbMATHGoogle Scholar
  37. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of Basic and Applied SciencesInternational Islamic UniversityIslamabadPakistan
  2. 2.Department of MathematicsIslamabad Model College for GirlsIslamabadPakistan

Personalised recommendations