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**On Kleene algebras and closed semirings.**
*(English)*
Zbl 0732.03047

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.]

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.

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 \({}^*\)-continuous. The categories of \({}^*\)-continuous Kleene algebras, closed semirings and S-algebras [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 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)].

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.

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 \({}^*\)-continuous. The categories of \({}^*\)-continuous Kleene algebras, closed semirings and S-algebras [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 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:

03G25 | Other algebras related to logic |

18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |

68Q70 | Algebraic theory of languages and automata |