Document Type : Original Manuscript


Department of Mathematics, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.


In this paper, the notion of fuzzy medial filters of a pseudo BE-algebra
is defined, and some of the properties are investigated. We show that the
set of all fuzzy medial filters of a pseudo BE-algebra is a complete lattice.
Moreover, we state that in commutative pseudo BE-algebras fuzzy filters and
fuzzy medial filters coincide. Finally, the notion of a fuzzy implicative filter is introduced
and proved that every fuzzy implicative filter is a fuzzy medial filter, and we
show that the converse is not valid in general.


1. A. Borumand Saeid, A. Rezaei and R. A. Borzooei, Some types of filters in BE-algebras, Math. Comput. Sci., 7(3) (2013), 341–352.
2. R.A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, On pseudo BE-algebras, Discuss. Math. Gen. Algebra Appl., 33 (2013), 95–108.

3. R.A. Borzooei, A. Borumand Saeid, A. Rezaei, A. Radfar and R. Ameri, On distributive pseudo BE-algebras, Fasciculi Math., 54 (2015), 21–39.

4. L.C. Ciungu, Commutative pseudo BE-algebras, Iran. J. Fuzzy Syst., 13(1) (2016), 131–144.

5. L.C. Ciungu, Commutative deductive systems of pseudo BCK-algebras, Soft Comput., 22(4) (2018), 1189–1201.

6. L.C. Ciungu, Fantastic deductive systems in probability theory on generalizations of fuzzy structures, Fuzzy Sets. Syst., 36(3) (2019), 113–137.

7. A. Di Nola, G. Georgescu and A. Iorgulescu, Pseudo BL-algebras, Part I, II. Multiple Val. Logic, 8 (2002), 673–714, 715–750.

8. G. Dymek, A. Walendziak, Fuzzy ideals of pseudo BCK-algebras, Demonstratio Math., XLV, 1 (2012), 1–15.

9. G. Georgescu and A. Iorgulescu, Pseudo MV-algebras, Multiple Val. Logic, 6 (2001), 95–135.

10. G. Georgescu and A. Iorgulescu, Pseudo-BCK algebras: an extension of BCKalgebras, Combinatorics, Computability and logic, 97–114, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 2001.

11. Y.B. Jun, H.S. Kim and J. Neggers, On pseudo-BCI ideals of pesudo-BCI algebras, Mate. Vesnik, 58(1-2) (2006), 39-46.

12. H.S. Kim and Y.H. Kim, On BE-algebras, Sci. Math. Jpn., 66(1) (2007), 113–117.

13. Y.H. Kim and K.S. So, On minimality in pseudo-BCI algebras, Commun. Korean
Math. Soc., 27(1) (2012), 7–13.

14. J. Rachu´┐┐ nek, A non commutative generalization of MV-algebras, Czehoslovak Math. J., 52 (127)(2) (2002), 255–273.

15. A. Rezaei and A. Borumand Saeid, On fuzzy subalgebras of BE-algebras, Afr. Mat., 22(2) (2011), 115–127.

16. A. Rezaei, A. Radfar and A. Pourabdollah, On medial filters of BE-algebras, Alg. Struc. Appl., 7(1) (2020), 127–141.

17. A. Walendziak and A. Rezaei, Fuzzy filters of pseudo BE-algebras, Afr. Mat., 31 (2019), 739–750.

18. A. Walendziak, M. Wojciechowska-Rysiawa, Fuzzy ideals of pseudo BCHalgebras, Math. Aeterna, 5(5) (2015), 867–881.

19. X.H. Zhang and Y.B. Jun, Solution of three open problems on pseudo filters (pseudo ideals) of pseudo BCK-algebras, 2010 International Conference on Artificial Intelligence and Computational Intelligence, (2011), 530–534.