Abstract
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, and that negation with the Kleene property (called the `Kleene negation') is describable by the set-theoretic complement. The propositional logic $${\mathcal {L}}_{K}$$LK of Kleene algebras is shown to be sound and complete with respect to a 3-valued and a rough set semantics. It is also established that Kleene negation can be considered as a modal operator, due to a perp semantics of $${\mathcal {L}}_{K}$$LK. Moreover, another representation of Kleene algebras is obtained in the class of complex algebras of compatibility frames. One concludes with the observation that $${\mathcal {L}}_{K}$$LK can be imparted semantics from different perspectives.
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