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On kleene algebras and closed semirings

  • Dexter Kozen
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 452)

Abstract

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].

Keywords

Transitive Closure Dynamic Logic Left Adjoint Forgetful Functor Multiplicative Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Dexter Kozen
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA

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