In this paper we consider fundamental properties of some types of filters
(implicative, positive implicative and fantastic filters) of non-commutative residuated
lattices and prove that every implicative filter and positive implicative filter are normal.
Moreover, we characterize some types of filters by the properties of corresponding
quotient algebras, that is, for any non-commutative residuated lattice X and a filter
F of X,
(a) F is an implicative filter if and only if X=F is a Heyting algebra;
(b) F is a positive implicative filter if and only if it is a Boolean filter if and
only if X=F is a Boolean algebra;
(c) If F is normal, then it is a fantastic filter if and only if X=F satisfies
the condition (a �� b) b = (b �� a) a and (a b) �� b = (b a) �� a
for all a; b �� X=F, that is, pseudo-MV algebra.