The Cube of Kleene Algebras and the Triangular Prism of Multirelations

  • Authors:

  • Koki Nishizawa;Norihiro Tsumagari;Hitoshi Furusawa

  • Affiliations:

  • Department of Information Systems, Faculty of Environmental and Information Studies, Tottori University of Environmental Studies,;Graduate School of Science and Engineering, Kagoshima University,;Department of Mathematics and Computer Science, Kagoshima University,

  • Venue:
  • RelMiCS '09/AKA '09 Proceedings of the 11th International Conference on Relational Methods in Computer Science and 6th International Conference on Applications of Kleene Algebra: Relations and Kleene Algebra in Computer Science
  • Year:
  • 2009

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Abstract

We refine and extend the known results that the set of ordinary binary relations forms a Kleene algebra, the set of up-closed multirelations forms a lazy Kleene algebra, the set of up-closed finite multirelations forms a monodic tree Kleene algebra, and the set of total up-closed finite multirelations forms a probabilistic Kleene algebra. For the refinement, we introduce a notion of type of multirelations. For each of eight classes of relaxation of Kleene algebra, we give a sufficient condition on type T so that the set of up-closed multirelations of T belongs to the class. Some of the conditions are not only sufficient, but also necessary.