On fuzzy filters of CI-algebras

In this paper after first introducing the fuzzy filter of CI-algebras we get several properties, we find that the notions of N-structured subalgebras, N-structured filters of CI-algebra and some of their basic properties and results are direct results on fuzzy filter of CI-algebra. Next we recall the contents of the filters in pseudo BL algebra. By exploring the equivalent condition of fuzzy normal filter, an open problem between two kinds of filters was solved. Jun and Xu et al. proved that in lattice implication algebra a positive implicative filter is implicative, and set the converse of it as an open problem. For the third part of this paper, we define the corresponding fuzzy filters and inspired by the fuzzy results we solve the open problem.

[1]  Lavinia Corina Ciungu.  Algebras on subintervals of pseudo-hoops , 2009, Fuzzy Sets Syst..

[2]  Zhang Xiao-hong.  On Fuzzy Filters and Fuzzy Ideals of BL-algebras , 2006 .

[3]  Sukhendu Kar,et al.  Fuzzy quasi-ideals and fuzzy bi-ideals of ternary semigroups , 2012 .

[4]  Yiquan Zhu,et al.  ON FUZZY IMPLICATIVE FILTERS OF LATTICE IMPLICATION ALGEBRAS , 2003 .

[5]  Xiao-hong Zhang,et al.  Pseudo-BL algebras and pseudo-effect algebras , 2008, Fuzzy Sets Syst..

[6]  N. P. Mukherjee,et al.  Fuzzy relations and fuzzy groups , 1985, Inf. Sci..

[7]  Y. Xu.  Lattice implication algebras , 1993 .

[8]  Esko Turunen,et al.  States on semi-divisible generalized residuated lattices reduce to states on MV-algebras , 2008, Fuzzy Sets Syst..

[9]  Arsham Borumand Saeid,et al.  Intuitionistic (T, S)-fuzzy CI-algebras , 2012, Comput. Math. Appl..

[10]  Petr Hájek.  Observations on non-commutative fuzzy logic , 2003, Soft Comput..

[11]  Esko Turunen,et al.  Boolean deductive systems of BL-algebras , 2001, Arch. Math. Log..

[12]  G. Dymek,et al.  Fuzzy filters of BE-algebras , 2013 .

[13]  Wei Wang,et al.  On fuzzy filters of pseudo BL-algebras , 2011, Fuzzy Sets Syst..

[14]  Li Kaitai,et al.  Boolean filters and positive implicative filters of residuated lattices , 2007 .

[15]  Li Kaitai,et al.  Fuzzy Boolean and positive implicative filters of BL-algebras , 2005 .

[16]  Y. Jun,et al.  Positive implicative and associative filters of lattice implication algebras , 1998 .

[17]  A. Rezaei,et al.  On fuzzy subalgebras of BE-algebras , 2011 .

[18]  Jianming Zhan,et al.  Interval valued $$(\in,\in\!\vee\,q)$$ -fuzzy filters of pseudo BL-algebras , 2008, Soft Comput..

[19]  Esfandiar Eslami,et al.  Some types of filters in BL algebras , 2006, Soft Comput..

[20]  K. Iseki,et al.  ON AXIOM SYSTEMS OF PROPOSITIONAL CALCULI XIV , 1966 .

[21]  Y. Jun.  Implicative filters of lattice implication algebras , 1997 .

[22]  Young Bae Jun,et al.  Fuzzy positive implicative and fuzzy associative filters of lattice implication algebras , 2001, Fuzzy Sets Syst..

[23]  Kaitai Li,et al.  Fuzzy filters of BL-algebras , 2005, Inf. Sci..

[24]  Chris Cornelis,et al.  Filters of residuated lattices and triangle algebras , 2010, Inf. Sci..

[25]  Kyung Ho Kim.  A Note on CI-Algebras , 2011 .

[26]  Kaitai Li,et al.  Fuzzy Boolean and positive implicative filters of BL-algebras , 2005, Fuzzy Sets Syst..

[27]  K. Sik,et al.  ON BE-ALGEBRAS , 2007 .