On kleene algebras and closed semirings
Kleene algebras are an important class of algebraic structures that arise in diverse areas of computer science: program logic and semantics, relational algebra, automata theory, and the design and analysis of algorithms. The literature contains at several inequivalent definitions of Kleene algebras and related algebraic structures [2,13,14,5,6,1,9,7].
•There is a Kleene algebra in the sense of  that is not *-continuous.
•The categories of *-continuous Kleene algebras [5,6], closed semirings [1,9] and S-algebras  are strongly related by adjunctions.
•The axioms of Kleene algebra in the sense of  are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103].
•Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt .
KeywordsTransitive Closure Dynamic Logic Left Adjoint Forgetful Functor Multiplicative Identity
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