On kleene algebras and closed semirings
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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].
•There is a Kleene algebra in the sense of  that is not *-continuous.
•The categories of *-continuous Kleene algebras [5,6], closed semirings [1,9] and S-algebras  are strongly related by adjunctions.
•The axioms of Kleene algebra in the sense of  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 .
KeywordsTransitive Closure Dynamic Logic Left Adjoint Forgetful Functor Multiplicative Identity
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- Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1975.Google Scholar
- John Horton Conway. Regular Algebra and Finite Machines. Chapman and Hall, London, 1971.Google Scholar
- John Horton Conway, personal communication, May 1990.Google Scholar
- Harel, D., First-Order Dynamic Logic. Lecture Notes in Computer Science 68, Springer-Verlag, 1979.Google Scholar
- Dexter Kozen, “On induction vs. *-continuity,” Proc. Workshop on Logics of Programs 1981, Spring-Verlag Lect. Notes in Comput. Sci. 131, ed. Kozen, 1981, 167–176.Google Scholar
- Dexter Kozen, “A completeness theorem for Kleene algebras and the algebra of regular events,” Cornell TR90-1123, May 1990.Google Scholar
- Werner Kuich, “The Kleene and Parikh Theorem in Complete Semirings,” in: Proc. 14th Colloq. Automata, Languages, and Programming, ed. Ottmann, Springer-Verlag Lecture Notes in Computer Science 267, 1987, 212–225.Google Scholar
- Saunders Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1971.Google Scholar
- Kurt Mehlhorn. Data Structures and Algorithms 2: Graph Algorithms and NP-Completeness. EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1984.Google Scholar
- K. C. Ng and A. Tarski, “Relation algebras with transitive closure,” Abstract 742-02-09, Notices Amer. Math. Soc. 24 (1977), A29–A30.Google Scholar
- K. C. Ng. Relation Algebras with Transitive Closure. PhD Thesis, University of California, Berkeley, 1984.Google Scholar
- Vaughan Pratt, “Semantical Considerations on Floyd-Hoare Logic,” Proc. 17th IEEE Symp. Found. Comput. Sci. 1976, 109–121.Google Scholar
- Vaughan Pratt, “Dynamic algebras and the nature of induction,” Proc. 12th ACM Symp. on Theory of Computing, 1980, 22–28.Google Scholar
- Vaughan Pratt, “Dynamic algebras as a well-behaved fragment of relation algebras,” in: D. Pigozzi, ed., Proc. Conf. on Algebra and Computer Science, Ames, Iowa, June 2–4, 1988; Spring-Verlag Lecture Notes in Computer Science, to appear.Google Scholar
- Arto Salomaa, “Two complete axiom systems for the algebra of regular events,” J. Assoc. Comput. Mach. 13:1 (January, 1966), 158–169.Google Scholar