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 several inequivalent definitions of Kleene algebras and related algebraic structures [2, 14, 15, 5, 6, 1, 10, 7]. In this paper we establish some new relationships among these structures. Our main results are: ffl There is a Kleene algebra in the sense of [6] that is not *-continuous. ffl The categories of *-continuous Kleene algebras [5, 6], closed semirings [1, 10] and S-algebras [2] are strongly related by adjunctions. ffl The axioms of Kleene algebra in the sense of [6] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [2, p. 103]. ffl Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt [15]. In Rov..