Special sets in a BL-algebra are introduced, and an example
in which a special set is not a deductive system is given.
Conditions for these special sets to be a deductive system are stated.
Using this special set, an equivalent condition of
a deductive system is established. We prove that a deductive system
can be represented by the union of such special sets.
The notion of commutative BL-algebras is also introduced,
and its characterizations are given.
The notion of sensitive deductive systems is introduced,
and some conditions for a deductive system
to be a sensitive deductive system are given.