  A structural theorem for Kleene algebras is proved, showing that an element
of a Kleene algebra can be looked upon as an ordered pair of sets. Further, we
show that negation with the Kleene property (called the `Kleene negation')
always arises from the set theoretic complement. The corresponding
propositional logic is then studied through a 3-valued and rough set semantics.
It is also established that Kleene negation can be considered as a modal
operator, and enables giving a perp semantics to the logic. One concludes with
the observation that all the semantics for this logic are equivalent.
