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Zbl 0732.03047
Kozen, Dexter
On Kleene algebras and closed semirings. (English)
[A] Mathematical foundations of computer science, Proc. 15th Symp., MFCS '90, Banská Bystrica/Czech. 1990, Lect. Notes Comput. Sci. 452, 26-47 (1990).

Summary: [For the entire collection see Zbl 0731.00026.] \par 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. \par Here we establish some new relationships among these structures. Our main results are: There is a Kleene algebra in the sense of the author's ``A completeness theorem for Kleene algebras and the algebra of regular events'' [Cornell TR 90-1123 (1990)] that is not ${}\sp*$-continuous. The categories of ${}\sp*$-continuous Kleene algebras, closed semirings and S-algebras [{\it J. H. Conway}, Regular algebra and finite machines (1971; Zbl 0231.94041)] are strongly related by adjunctions. The axioms of Kleene algebra in the sense of the author [loc. cit.] are not complete for the universal Horn theory of the regular events. This refutes a conjecture of Conway [loc. cit.]. Right-handed Kleene algebras are not necessarily left-handed Kleene algebras. This verifies a weaker version of a conjecture of {\it V. Pratt} [``Dynamic algebras as a well-behaved fragment of relation algebras'', Proc. Conf. Algebra and Comput. Sci., Ames, Iowa 1988, Lect. Notes Comput. Sci. (to appear)].

MSC 2000:: *03G25 Other algebras related to logic
18A40 Adjoint functors
68Q70 Algebraic theory of automata

Keywords: Kleene algebras; ${}\sp *$-continuous Kleene algebras; closed semirings; S-algebras; regular events

Citations: Zbl 0731.00026; Zbl 0231.94041

Cited in: Zbl 0814.03024

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