Abstract:

The structure of non-commutative residuated lattices is further studied.With the help of the thought and method of fuzzy mathematics, the concept of fuzzy filters is introduced, and the relations between fuzzy filters and crisp filters of non-commutative residuated lattices are discussed. On the basis of fuzzy filters, fuzzy implicative filters and fuzzy positive implicative filters of non-commutative residuated lattices are introduced. The basic properties of them are discussed, and then several equivalent characterizations for fuzzy implicative filters and fuzzy positive implicative filters are given.It is proved that fuzzy positive implicative filters must be fuzzy implicative filters. Under certain conditions, the equivalence of fuzzy implicative filters and fuzzy positive implicative filters is proved.

Key words: fuzzy logic; non-commutative residuated lattices; fuzzy filters; fuzzy implicative filters; fuzzy positive implicative filters