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

  • Tahir Mahmood
  • Muhammad Irfan Ali
  • Azmat Hussain
Article
First Online:
Received:
Revised:
Accepted:
  • 49 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

08A72 34C41 06F05 
This is a preview of subscription content, log in to check access.

References

  1. Banerjee M, Pal SK (1996) Roughness of a fuzzy set. Inf Sci 93:235–246MathSciNetCrossRefMATHGoogle Scholar
  2. Bhakat SK, Das P (1996) \(\left( \in,\in \vee q\right) \)-fuzzy subgroups. Fuzzy Set Syst 80:359–368CrossRefMATHGoogle Scholar
  3. Biswas R, Nanda S (1994) Rough groups and rough subgroups. Bull Polish Acad Sci Math 42:251MathSciNetMATHGoogle Scholar
  4. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in rough set theory. Inf Sci 107:149–167MathSciNetCrossRefMATHGoogle Scholar
  5. Chakrabarty K, Biswas R, Nanda S (2000) Fuzziness in rough sets. Fuzzy Sets Syst 110:247–251MathSciNetCrossRefMATHGoogle Scholar
  6. Davvaz B (2004) Roughness in rings. Inf Sci 164:147–163MathSciNetCrossRefMATHGoogle Scholar
  7. Davvaz B (2008) A short note on algebra T-rough sets. Inf Sci 178:3247–3252MathSciNetCrossRefMATHGoogle 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 176:3658–3674MathSciNetCrossRefMATHGoogle Scholar
  10. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–208CrossRefMATHGoogle 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 7:337–350MathSciNetMATHGoogle Scholar
  12. Hosseini SB, Jafarzadeh N, Gholami A (2012) T-rough ideal and T-rough fuzzy ideal in a semigroup. Adv Mater Res 433:4915–4919CrossRefMATHGoogle Scholar
  13. Iwinski TB (1987) Algebraic approach to rough sets. Bull Polish Acad 35:673–683MathSciNetMATHGoogle Scholar
  14. Jirojkul C, Chinram R (2009) Fuzzy quasi-ideal subsets and fuzzy quasi-filters of ordered semigroup. Int J Pure Appl Math 52:611–617MathSciNetMATHGoogle Scholar
  15. Jun YB (2003) Roughness of ideals in BCK-algebra. Sci Math Jpn 57:165–169MathSciNetMATHGoogle 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 12:1119–1124CrossRefMATHGoogle 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 178:1343–1354MathSciNetCrossRefMATHGoogle Scholar
  19. Kehayopulu N, Tsingelis M (1999) A note on fuzzy sets in semigroups. Sci Math 2:411–413MathSciNetMATHGoogle Scholar
  20. Kehayopulu N, Tsingelis M (2002) Fuzzy sets in ordered groupoids. Semigroup Forum 65:128–132MathSciNetCrossRefMATHGoogle Scholar
  21. Kehayopulu N, Tsingelis M (2005) Fuzzy bi-ideals in ordered semigroups. Inf Sci 171:13–28MathSciNetCrossRefMATHGoogle Scholar
  22. Kehayopulu N, Tsingelis M (2006) Fuzzy interior ideals in ordered semigroups. Lobachevskii J Math 21:65–71MathSciNetMATHGoogle Scholar
  23. Kehayopulu N, Tsingelis M (2009) Fuzzy right, left, quasi-ideals, bi-ideals in ordered semigroups. Lobachevskii J Math 30:17–22MathSciNetCrossRefMATHGoogle Scholar
  24. Kuroki N (1979) Fuzzy bi-ideals in semigroups. Commentarii Mathematici Universitatis Sancti Pauli 28:17–21MathSciNetMATHGoogle Scholar
  25. Kuroki N (1981) On fuzzy ideals and fuzzy bi-ideals in semigroups. Fuzzy Sets Syst 5:203–215MathSciNetCrossRefMATHGoogle Scholar
  26. Kuroki N (1991) On fuzzy semigroups. Inf Sci 53:203–236CrossRefMATHGoogle Scholar
  27. Kuroki N (1993) Fuzzy semiprime quasi-ideals in semigroups. Inf Sci 75:201–211MathSciNetCrossRefMATHGoogle Scholar
  28. Kuroki N (1997) Rough ideals in semigroups. Inf Sci 100:139–163MathSciNetCrossRefMATHGoogle Scholar
  29. Ma X, Zhan J, Jun YB (2009) On \((\in,\in \vee q)\) fuzzy filters of \(R_{0}\)-algebras. Math Logic Quart 55:493–508CrossRefMATHGoogle 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 11:341–356MATHGoogle Scholar
  32. Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35:512–517MathSciNetCrossRefMATHGoogle 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 11:371–382MathSciNetMATHGoogle Scholar
  35. Yao YY (1998) Constructive and algebraic methods of the theory of rough sets. Inf Sci 109:21–47MathSciNetCrossRefMATHGoogle Scholar
  36. Yuan X, Zhang C, Ren Y (2003) Generalized fuzzy groups and many-valued implications. Fuzzy Sets Syst 138:205–211MathSciNetCrossRefMATHGoogle Scholar
  37. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle 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