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

Kleene Algebras and Logic: Boolean and Rough Set Representations, 3-valued, Rough and Perp Semantics