Title: New types of bipolar fuzzy ideals of BCK-algebras
Author(s): G. Muhiuddin, D. Al-Kadi, A. Mahboob, K.P. Shum
Pages: 859-875
Cite as:
G. Muhiuddin, D. Al-Kadi, A. Mahboob, K.P. Shum, New types of bipolar fuzzy ideals of BCK-algebras, Int. J. Anal. Appl., 18 (5) (2020), 859-875.


The notions of bipolar fuzzy closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are introduced, and related properties are investigated. Characterizations of a closed, bipolar fuzzy positive implicative, bipolar fuzzy implicative ideals of BCK-algebras are given, and several properties are discussed. Finally, we prove that if T is an implicative BCK-algebra, then a fuzzy subset µ of T is a bipolar fuzzy ideal of T if and only if it is a bipolar fuzzy implicative ideal of T.

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  1. D. Al-Kadi, G. Muhiuddin, Bipolar fuzzy BCI-implicative ideals of BCI-algebras, Ann. Commun. Math. 3 (1) (2020), 88–96. Google Scholar

  2. A. Al-Masarwah, A.G. Ahmad, On some properties of doubt bipolar fuzzy H-ideals in BCK/BCI-algebras, Eur. J. Pure Appl. Math. 11 (2018), 652-670. Google Scholar

  3. Anas Al-masarwah, Abd Ghafur Ahmad, G. Muhiuddin, Doubt N-ideals theory in BCK-algebras based on N-structures, Ann. Commun. Math. 3 (1) (2020), 54–62. Google Scholar

  4. A.M. Al-roqi, G. Muhiuddin, S. Aldhafeeri, Normal Unisoft Filters in R0-algebras. Cogent Math. 4 (2017), 1383006. Google Scholar

  5. J. Chen, S. Li, S. Ma, X. Wang, m-polar fuzzy sets: An extension of bipolar fuzzy sets, Sci. World J. 2014 (2014), Article Id 416530. Google Scholar

  6. P.A. Ejegwa, J. A. Otuwe, Frattini fuzzy subgroups of fuzzy groups Ann. Commun. Math. 2 (1) (2019), 24-31 Google Scholar

  7. K. Hayat, T. Mahmood, B.Y. Cao, On bipolar anti fuzzy H-ideals in hemirings, Fuzzy Inf. Eng. 9 (2017), 1–19. Google Scholar

  8. M. Ibrara, A. Khana, B. Davvazb, Characterizations of regular ordered semigroups in terms of (α, β)-bipolar fuzzy generalized bi-ideals, J. Intell. Fuzzy Syst. 33 (2017), 365-376. Google Scholar

  9. Y. Imai, K. Iseki, On axiom systems of propositional calculi, Proc. Japan Acad. 42 (1966), 19-22. Google Scholar

  10. K. Is´eki, An algebra related with a propositional calculus, Proc. Japan Acad. 42 (1966) 26-29. Google Scholar

  11. C. Jana, M. Pal, A.B. Saeid, (∈, ∈ ∨q)-Bipolar fuzzy BCK/BCI-algebras, Missouri J. Math. Sci. 29 (2017), 139-160. Google Scholar

  12. Y.B. Jun, M.S. Kang, H.S. Kim, Bipolar fuzzy hyper BCK-ideals in hyper BCK-algebras, Iran. J. Fuzzy Syst. 8 (2) (2011), 105–120. Google Scholar

  13. Y.B. Jun, Closed fuzzy ideals in BCI-algebras, Math. Japon. 38 (1) (1993), 199-202. Google Scholar

  14. K. Kawila, C. Udomsetchai, A. Iampan, Bipolar fuzzy UP-algebras, Math. Comput. Appl. 23 (4) (2018), 69. Google Scholar

  15. K.J. Lee, Bipolar fuzzy subalgebras and bipolar fuzzy ideals of BCK/BCI-algebras, Bull. Malays. Math. Sci. Soc. 32 (2009), 361–373. Google Scholar

  16. K.M. Lee, Bipolar-valued fuzzy sets and their operations, Proc. Int. Conf. Intell. Technol. Bangkok, Thailand (2000), 307–312. Google Scholar

  17. K.M. Lee, Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets, J. Fuzzy Log. Intell. Syst. 14 (2004), 125–129. Google Scholar

  18. J. Meng, Y.B. Jun, H. S. Kim, Fuzzy implicative ideals of BCK-algebras, Fuzzy Sets Syst. 89 (1997), 243-248. Google Scholar

  19. J. Meng, Y.B. Jun, BCK-algebras, Kyung Moon Sa Co., Seoul Korea 1994. Google Scholar

  20. G. Muhiuddin, Habib Harizavi, Y.B. Jun, Bipolar-valued fuzzy soft hyper BCK ideals in hyper BCK algebras, Discrete Math. Algorithms Appl. 12 (2) (2020), 2050018. Google Scholar

  21. G. Muhiuddin, Bipolar fuzzy KU-subalgebras/ideals of KU-algebras, Ann. Fuzzy Math. Inform. 8, (3) (2014), 409-418. Google Scholar

  22. G. Muhiuddin, K. P. Shum, New types of (α, β) - fuzzy subalgebras of BCK/BCI-algebras, Int. J. Math. Computer Sci. 14 (2) (2019), 449–464. Google Scholar

  23. G. Muhiuddin, S. Aldhafeeri, Characteristic fuzzy sets and conditional fuzzy subalgebras, J. Comput. Anal. Appl. 25 (8) (2018), 1398–1409. Google Scholar

  24. G. Muhiuddin, A.M. Al-Roqi, Classifications of (alpha, beta)-fuzzy ideals in BCK/BCI–algebras, J. Math. Anal. 7 (6) (2016), 75–82. Google Scholar

  25. G. Muhiuddin, A.M. Al-Roqi, Subalgebras of BCK/BCI-algebras based on (α, β)-type fuzzy sets, J. Comput. Anal. Appl. 18 (6) (2015), 1057–1064. Google Scholar

  26. T. Senapati, K.P. Shum, Cubic subalgebras of BCH-algebras, Ann. Commun. Math. 1 (1) (2018), 65-73. Google Scholar

  27. A.F. Talee, M.Y. Abbasi, A. Basar, On properties of hesitant fuzzy ideals in semigroups, Ann. Commun. Math. 3 (1) (2020), 97-106 Google Scholar

  28. S. Thongarsa, P. Burandate, A. Iampan, Some operations of fuzzy sets in UP-algebras with respect to a triangular norm, Ann. Commun. Math. 2 (1) (2019), 1-10 Google Scholar

  29. L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965), 338-353. Google Scholar

  30. W.R. Zhang, Bipolar fuzzy sets, Proc. Fuzzy-IEEE (1998), 835-840. Google Scholar


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