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: (1) There is a Kleene algebra in the sense of [6] that is not *-continuous. (2) The categories of *-continuous Kleene algebras [5, 6], closed semirings [1, 9] and S-algebras [2] are strongly related by adjunctions. (3) 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]. (4) Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of Pratt [14].