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]. In this paper we establish some new relationships among these structures. Our main results are:•There is a Kleene algebra in the sense of [6] that is not *-continuous.•The categories of *-continuous Kleene algebras [5,6], closed semirings [1,9] and S-algebras [2] are strongly related by adjunctions.•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].•Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt [14].